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ONTARIO HIGH SCHOOL ALGEBRA AUTHORIZED BY THE MINISTER OF EDUCATION Storage - Item . EDUCATION %H COMPANY OF CANADA LIMITED LP6 - J35A UBC Library RONTO C5am f^ iz ^ Digitized by the Internet Archive in 2010 with funding from University of British Columbia Library http://www.archive.org/details/highschoolalgebOOcraw HIGH SCHOOL ALGEBBA HIGH SCHOOL ALGEBRA^'' BY J. T. CRAWFORD, B.A. chief instructor ix mathematics, university schools, professor of mathematics, ontario college of education t:niversity of Toronto REVISED EDITION TORONTO The macmillan company of Canada, limited Copyright. Canada, ini« By THE iL\CMILLAX COMPANY OF CANADA LIMITED PRINTED IX CANADA PREFACE This text covers the work prescribed for entrance to the Universities and Normal Schools. The book is written from the standpoint of the pupil, and in such a form that he will be able to understand it with a minimum of assistance from the teacher. The question method is frequently used in developing the theory. The purpose of this is to lead the pupil to think for himself. The close connection between algebra and arithmetic is constantly kept in view, and in manv cases the arithmetical and algebraic processes are shown in parallel columns. There are numerous diagrams for the purpose of illus- trating the theory, and algebraic methods are applied to many of the theorems which the pupil meets in elementary geometry. Special emphasis is placed upon the verification of results. In the past, sufficient attention has not been given to this important part of mathematical work. Provision is made for oral work, many of the exercises being introduced by a number of oral examples for use in class. The equation and the solution of simple problems are 'jitroduced in the second Chapter. It is hoped that the pupil will thus become interested much earlier in the work. Long multiplications and divisions are not included in the work of the first year. They are difficult for the beginner and of little interest, as there is not much to offer in the way of practical illustrations. VI PREFACE Chapter X., with which the pupil would begin the second year's work, contains a thorough review of the simple rules. Here the more complicated processes are dealt with. The graphical work is introduced naturally in illustrating the negative quantity and in the solution of equations. Only graphs which can be drawn with the ruler and compasses are included in the book. More attention is given to methods of inspection in the extraction of roots. The long process for cube root is eUminated, as cube root is not now required in arithmetic. The work on ratio and proportion is presented in as simple a form as possible, and is intended onl}' as an introduction to the senior work in this subject. The geometrical illustrations which are given should make it more interesting. The division method of finding highest common factor has been discarded, as it is usually performed mechanically and not understood by pupils. The elimination method which is used will be found easy to apply with expressions which are not too comphcated. Finding the highest common factor of expressions of the fourth or higher degrees is of little algebraic value, and few examples of such problems will be found in the book. The review exercises at the end of each Chapter will be found useful, particularly for the purpose of reviewing the work of a previous term. On the recommendation of experienced teachers the answers are not given to simple examples, or to such examples as the pupil can verify without difficulty. CONTENTS CHAPTER I PABV Algebraic Notation 1 Arithmetical and Algebraic Signs and Symbols (1). Fundamental Laws (3). Factor and Pro- duct (6). Power and Index (7). Terms (8). Coefficient (9). Addition and Subtraction of like Terms (9). Use of Brackets (11). Review Exercise (13). CHAPTER II Simple Equations ....:.... 16 Meaning of Equation, Solving an Equation, Root of an Equation (16). Axioms used (18). Veri- fying Results (19). Problems solved by Equations (21). Review Exercise (25). CHAPTER III Positive and Negative Numbers 28 Graphical Representation of Positive and Negative Numbers (28). Concrete Examples of Negative Numbers (30). Signs of Operation and Signs of Quahty (32). Absolute Value (32). Review Exercise (34). CHAPTER IV Addition and Subtraction 36 Addition of Quantities with like Signs (36). Com- pound Addition (37). Addition with unlike Signs (39). Indicated Additions (41). Sub- traction the Inverse of Addition (42). Rule for Subtraction (43). Removal of Brackets (45). Review Exercise (47). CONTENTS CHAPTER V Multiplication and Division 49 Multiplication of Simple Positive Quantities (49). Index Law for Multiplication (49). Rule of Signs (50). Compound Multiplication (53). Verifications (55). Division by a Simple Quantity (57). Index Law for Division (58). Review Exercise (60). CHAPTER VI Simple Equations {continued) 62 Equation and Identity' (62). Transposing Terms (63). Simple Fractional Equations (67). Steps in the Solution of an Equation (67). Problems (69). Algebraic statements of Arithmetical Theorems (74). Review Exercise (75). CHAPTER VII Simultaneous Equations 78 Equations with two Unknowns (78). Method of Solution (79). Ehmination (80). Fractional Simultaneous Equations (82). Problems (83). Review Exercise (85). CHAPTER Vin Type Products and Simple Factoring 88 Monomial Factors (88). Product of two Binomials (89). Factors of Trinomials (90). Radical Sign (92). Square of a Binomial (94). Square Root of a Trinomial (95). Difference of two Squares (97). Numerical Applications (99). Review Exercise (101). CONTENTa ti CHAPTER IX Simple Applications of Factoring ..... 10? Highest Common Factor (103). Algebraic Fractions (104). Lowest Terms (105). Multiplication and Division of Fractions (106). Lowest Common Multiple (107) Addition and Subtraction of Fractions (108). Mixed Expressions (110). Review Exercise (111). CHAPTER X Review of the Simple Rules 114 Brackets (114). Collecting Coefficients (117). Multiplication with Detached Coefficients (118). Division by a Compound Quantity (121). Veri- fying Division (122) Inexact Division (124). Review Exercise (125). CHAPTER XI Factoring (continued) 128 Factors by Grouping (128). Complete Squares (130). DifEerence of Squares (132). Incomplete Squares (135). Trinomials (137), Sum. and Differenc«» of Cubes (140). The Factor Theorem (141). Easy Quadratic Equations 1144). Review Exercise (146). CHAPTER Xn Simultaneous Equations {continued) ... 149 Elimination by Substitution and by Comparison (149). Equations with three Unknowns (152). Special Forms of Equations (154). Solution of Problerj>» '156). Review Exercise (161). X CONTENTS CHAPTER Xm PAOX Geometrical Representation of Number .... 165 Function of x (165). Variables and Constants (165). Arithmetical Graphs (166). The Axes (169). Graph of an Equation (170). Coordinates (170). Plotting Points (171). Linear Equation (173). Graphical Solution of Simultaneous Equations (175). Special Forms of Equations (176). Review Exercise (177). CHAPTER XIV Highest Common Factor and Lowest Common Multiple 180 By Factoring (180). By Elimination (183). Product of the H.C.F. and L.C.M. (185). Review Exercise (187). CHAPTER XV Fractions .... 188 Changes in the Form of a Fraction (188). Lowest Terms (190). Addition and Subtraction (191). Special Types (193). Cyclic Order (194). Multiplication and Division (197). Complex Fractions (199). Review Exercise (201). CHAPTER XVI Fractional Equations ... 204 Cross Multiplication (204). Special Forms of Fractional Equations (206). Literal Equations with one Unknown (209), with two Unknowns (212). Review Exercise (214). COlfTENTS Jd CHAPTER XVn PAGE Extraction of Roots 216 Square Root by Inspection (216), by the Formal Method (217). Verifying Results (218). Cube of a Binomial (222). Cube Root (223). Higher Roots (224). Review Exercise (226). CHAPTER XVIII Quadratic Surds 228 Surd and Rational Quantities (228). Mixed and Entire Surds (229). Like Surds (23D. Addition and Subtraction (231). Square Root Table (232). Conjugate Surds (233). Rationalizing a De- nominator (235). Surd Equations (237). Review Exercise (238). CHAPTER XIX Quadratic Equations 240 Standard Form (241). Solution by Factoring (242), by Completing the Square (244). Irrational Roots (247). Inadmissible Solutions (248). Review Exercise (250). CHAPTER XX Ratio and Proportion 253 Methods of Comparing Magnitudes (253). Com- parison of Ratios (254). Proportion (256). Finding a Ratio by Solving an Equation (257). Mean Proportional (261). Ratio Theorems (263). Review Exercise (266). :m CONTENTS CHAPTER XXT PAOI The General Quadratic Equation 268 Solution of Literal Quadratics (268). Solution by Formula (270). Imaginary Roots (271). Equa- tions Solved like Quadratics (273). Review Exercise (276). CHAPTER XXII Simultaneous Quadratics 279 Three Types of Simultaneous Quadratics (279/. Special Methods (284). Graphical Solutions f288). Review Exercise (290). CHAPTER XXIII Indices 293 The Index Laws (294). Fractional, Zero and Negative Indices (297). Operations with Fractional and Negative Indices (301). Con- tracted Methods (303). Review Exercise (305). CHAPTER XXIV Surds and Surd Equations 308 Surds of different Orders (308). Varying Forms of Surds (309). Surd Equations (312). Extraneous Roots (313). Square Root of a Binomial Surd (317). Imaginary Surds (320). Impossible Problems (323). Review Exercise (324). CHAPTER XXV Theory of Quadratic Equations 327 Sum and Product of the Roots (327). Reciprocal Roots (328). Functions of the Roots (331). Character of the Roots (335). The Discriminant (337). Factors of the Quadratic Expression (338). Review Exercise (341). CONTENTS CHAPTER XXVI Supplementary Chapter 344 Factors of the Product of two Trinomials (344). Sum and Difference of Cubes (346). Factors by Grouping (349). The Factor Theorem (351), Symmetry (354). Factoring by Sym- metry (355). Identities (359). Review Exercise f362). Answers 365 HIGH SCHOOL ALGEBRA HIGH SCHOOL ALGEBRA CHAPTER I ALGEBRAIC NOTATION I. Use of Arithmetical Signs. In arithmetic, signs are used to abbreviate the work. In algebra the same signs are used, with the same meanings and for the same purpose. BZSRCISB 1 Write tiie following statements in the shortest way you can, using the signs and symbols with which you are familiar in arithmetic. 1. Two and two make four. 2. The sum of five, ten and twentj' is thirty-five. 3. Six and four is the same as four and six. 4. Seven times eight is the same as eight times seven. 5. The difference between twelve and five is seven. 6. Ten exceeds six by four. 7. The excess of twenty over fifteen is five. 8. The defect of thirty from a hundred is seventy. 9. Thirty-six divided by four is nine. 10. Three score and ten is seventy. II. One half of the sum of seven and five is six. 12. The sum or the product of three, five and seven is the same in whatever order they are written. 13. Three multipUed by four is twelve, therefore twelve divided by three is four. 14. The square of four is sixteen, therefore the square root of sixteen is four. B 2. Algebraic Symbols. In the preceding exercise you have used symbols to represent the numbers stated and signs tc show the operations performed on those numbers. In algebra, sj-mbols are used more extensively than ic Arithmetic. A B I H If the length of this line be measured it will be found tc be two inches. But without measuring it, we may say that the measure of its length is some definite number which might be represented by the letter a. The measure of the length of another line might be repre- sented by h. The cost of an article might be c cents, or the cost of a farm might be x dollars, or the weight of a stone might be m pounds. Here a, b, c, x, m are algebraic number-symbols, or briefly algebraic numbers. The symbols 1, 2, 3, etc., used to represent numbers in arithmetic are called arithmetical number-symbols or arith- metical numbers. In algebra the number symbols of arithmetic are also used. For the present, when letters are used to represent numbers, t will be understood that each letter represents some integral or fractional number. 3. Signs of Multiplication. In this square the measure of D C the length of the side AB is a. What is the measure of the length of BC ; of CD ; of AB-}-BC ; of AB-{-BC-\-CD 1 The measure of the perimeter (sum of all the sides) is a+a-j-a+a or 4 times a or 4xa. In algebra, ' xa or a X 4 is usually written 4a, the sign oi multipUcati a being understood. It is also written 4 . a, the dot repreijnting multipHcation. Thus, 4xa=4.a=4a, and as in arithmetic, is a short way of writing a+a+a+a. Thus, if a =6, the measurf of the perimeter cf the squar? ie »+6-|-6+6=4xt = M, ALOBBRAIC NOTATION S ft will be observed that in algebra the multiplication of a and 4 is only indicated in the form 4a, while in arithmetic it may be actually performed as in the result 24. The pupil must recognize the difference between 24 {twenty-four) and the product of 2 and 4 or 2 x 4 or 2 . 4. When two numerical quantities are to be multiplied, the sign of multiplication must be used, so that as stated, 24 may be distinguished from 2x4. When both factors are not numerical as 4xa or axb, the sign is omitted and these are written in the form 4a, ah. 4. Signs of Division. As in arithmetic, the quotient ob- tained by dividing one number by another may be TVTitten in the fractional form. In arithmetic the division may be actualty performed, as in 6-T-3, which may be written | or 2, but it is frequently only indicated as in 6-^7, which is written 5. So in algebra, the quotient obtained on dividing a by h, or a-i-6, is written -r, and here, as in multipUcation, the division can only be indicated unless the numerical values of a and h are known. 5. Some Fundamental Laws. Since the letters used in algebra represent arithmetical numbers, all the laws of arithmetic must be true also in algebra. In arithmetic. \ In algebra. (1) 7 + 3 = 3 + 7. ' ' (1) a+6 = 6+a. 6+2+5 = 6+5+2 = 2+5+6. 0+6+C = a-\-c-{-b=b-\-c-\-a. (2) 3x5 = 5x3. (2) ah = ba. 2x4x3 = 2x3x4 = 3x4x2. abc = acb = cba. (3) 10 + 5-2=10-2 + 5. (3) a + 6 — c=a — c + 6. 10-5-2 = 10-2-5. a— 6— c=o— c — 6. (4)3x 10 + 5 = 3 + 5x10=10 + 5x3 (4) a x6 + c = a+cx6=6 + cxa. From (1) and (2) it follows that the sum or the product of several numbers is independent of th^ order in which they are written. ^2 6 ALGEBRA From (3) and (4) it follows that a series of additions and subtractions, or of multiplications and divisions, may be made in any order. In finding the numerical value of an expression hke 3a4-46— 2c for given values of a, b and c, the operations are performed in the same order as in arithmetic, the multi- phcations being performed first and then the additions and subtractions in any order Thus, when a = 2, 6 = 3, c=l, 3a+46-2c=3x 2+4x3-2x1 = 6+12-2=16. Similarly, for the same values of a, b, c, ab+bc _ 2x3 + 3x 1 _ 6+3 _ 9 a+b " 2 + 3 — 5 ~5* Note. — Many of the examples in the following exercise may be taken orally. The pupil, however, is advised to write the algebraic forms so that he may thereby become familiar with themi. EXSROISB 2 1. When a = 6, what are the numerical value? of: _ 1 o 5 12 2 5a., 6a, -a, -, —a, — , , —-7 2 3 6 a a 3 2. When x=5 and j/=3, what are the values of : x+y, x—y, xy, 3x+2i/, 2x—3y, Ixy ? 3. When m—4i, n=6, r=2, find the values of ; m-\-n-\-r, m-{-r--n, mn-{-mr, mr—n, 4n— 3to— 6r. 4. Express algebraically the sum, the difference and the product of a and h. What are their values when a=8 and fc=3 ? 5. The quantities a, b and c are to be added together. Express the sum algebraically. What is its value when a=6, 6=4, c=12 ! 6. When a is divided by 6 the quotient is expressed in the form -. When c is added to the quotient of x by y, how is the result expressed t What is its value when z=12, ?/=4, c=10 ' 7. A boy has p marbles ; he wins q marbles and then loses r marbl***. How many has he now ? How many if p=5, q—\\, r=4 ? ALOBBRAIC NOTATION 6 8, When o=4 and 6=5, find the numerical value of 12a-56 + 6o-76-i-10, 9. The sides of a triangle are a, b and c ; express algebraically the perimeter and tlie semi-perimeter. What do they become if a=13, 6=14, c=15? 10. Find the cost of 8 articles at 5 cents each ; of 7 articles at k cents each ; of x yards of cloth at b cents a yard ; of m tons of coal at n dollars a ton. 11. How many cents are there in 4 dollars ; in x dollars ; in x dollars and y cents ; in a quarters and b ten-cent pieces ? 12. Find the number of inches in 2 yards ; in 3 feet and 7 inches ; in a yards ; in 6 feet ; in z feet and y inches ; in m yards n feet and V inches. 13. What operations are to be performed to find the numerical ealue of ma-\-nb, when a =2, 5=5, m=3, n=6 ? What is the value ? 14. What operations are to be performed to find the value of -^^^ a-\-b when a =5, 6=6, x=15, t/=7 ? What is the value ? 15. By varying the order of the letters, in how many ways can you write a+b-^c ? 16. In how many different ways can you write zyz f 17. In the figure, BC is twice as long as ^5. If A B C AB \& I units in length, what is the length of BC 1 ' ' of AC ? 18. In the figure, BC is three times as long as AB and CD is twice as long as AB. If AB is x units in /\ b C D length, what are the lengths of BC ? ' ^ 1 ' CDt BDl AD1 19. In the following statements c represents the cost of an article, s the selling price, and g the gain : (1) s-c=g, (2) c-\-g=s, (3) s—g=c. Read them and explain their meanings. 20. What is the next integer above 27 ? The next below 27 7 What is the next integer above n ? The next below n 1 21. If n is an even integer, what is the next even integer above it and the next even inteser below it ? « ALGEBRA 22. If I is any number,, what is the number which is 5 greater than z ? 5 less than x ? 23. A boy is 10 years old. How old will he be in 6 years ? In m years ? How old was he 4 years ago ? n years ago ? 24. A man is x years old. How old will he be in n years ? How old was he m years ago ? In how many years will he be three times as old as he is now ? 25. A boy wae p years old 3 years ago. How old will he be 15 years from now ? 26. Explain the difference between , a+- and a.-. What c c c are their values when a=6, 6—9, c=3 ? 27. The side of one square is a and of a smaller one is 6. Indicate the difference in their perimeters. What Is the difference if a=10 and 6=6? 28. The sides of one rectangle are a and b, and of another are c and d. Indicate the difference in their areas, (1) when the first rect- angle is the larger, (2) when the second is the larger. 29. What arithmetical number does 10x4-?/ represent when x=5 V=3 ? When x^l, y=^ ? 30. When a=3, 6=4, c=5, d=0, find the values of : (1) 10a+46-5c+3d. (2) 5ab+2cd-3ac. (3) \ac+^c-lad. (4) -r-. 2o-t-6— c+a 6. Factor and Product. WTien numbers are multiplied together the result is called the product, and the numbers which were multipUed are called the factors of the product. Thus, 3x5=15, therefore the feictors of 15 are 3 and 5, so a xb = ab, therefore the factors of ab are a and b. The factors of 3z are 3 and x. The 3 is called a numerical factor and the x, a literal factor. Just as 12 may have different sets of factors as 3x4, 2x6, 2 X 2 X 3, so 3xy has the factors Sxxy,3xxy, xxZy or Sxxxy. The prime factors of 12 are 2, 2 and 3. and the simplest factors of 3ry are 3. x and y. ALGEBRAIC NOTATION 9 In whatever order the factors are written the product is the same, but it is usual to write the numerical factor first and the literal factors in alphabetical order. 7. Power and Index. What is the area of a square whose side is 7 inches in length ? The measure of the area of the- square in art. 3 is axa, which is written a^, and is read " a square," or " a to the second power." The product when 2 a's are multiphed together is called the power, and the 2 is called the index or exponent of the power. If the edge of a cube is 6 inches, what is the sum of all the edges ? What is the area of each face of the cube ? What is the area of all the faces ? What is the volume of the cube ? If the edge of a cube is a, the sum of all the edges is 12a. The area of each face is a^, and of all the faces is Ga^. The volume is axaxa or a^, which is read "a cube," or " a to the third power." The pupil must distinguish between 3a and a^. The former means 3xa, and the latter axaxa. Thu8,ifa = 5, 3o=3x5=15, but 0^ = 5x5x5= 125, BXERCISE 3 (1-14, Oral) 1. What are the prime factors of 35, of 42, of 75 ? 2. What are the simplest factors of 5xy, of 6mn ? 3. Express 3a6c as the product of two factors in four different ways. 4. Give two common factors of 15a6 and 256c. 5. Find the values of 3^ x 2^, 10^ x 5\ 2* x5,3^x2^x 5. 6. Using an index, express 100 as a power of 10, 16 as a power oJ 2, 27 as a power of 3, 625 as a power of 5. 7. What is a short way of writing a+al a+a+al axal axaxat aaaa 1 8. What is the area of a square whose side is 6 inches ? whose side is x inches 7 B AW)SBRA 9. What is the volume of a cube whose edge is 3 inches ? whose edge is m inches ? 10. When a=4, what is the value of a^ ? of 2a ? What is theii difference ? 11. When x=2, what is the difference b^iween x^ and 3x ? 12. What is the difference between "x square-' and "twice x" when x=ll ? 13. If Tn=\0, what is the difference between the square of 3m and three times the square of m ? 14. The side of a square is x inches and of a smaller one is y inches. What is the sum of their areas ? What is the difference ? What do these results become when x— 10 and y=6 ? 15.* If x=6 and y=2, find the numerical values of 3x2, x^+y, x+y-, x^-y\ 2x^-3y^. 16. Find the values of x'+x^-f-x for the following values of x ; x=l, 2, 3, 0. 17. If j/=4x-— 7, find the value of y if x=2, if x=3, if x=2^. 18. The unit of work is " a day's work, ' that is, the work which one man can do in one daj. How many units of work can 3 men do in 5 days ? 6 men in x days ? m men in n days ? a men in a days T 19. If a=3, 6=2, c=l, find the quotient when a^-\-b^-}-c^ is divided by 2a+b — c. 20. Show that x^+26x has the same value as 9x^-1-24 when x=2 or 3 or 4. 21. If x=10 and y=5, how much greater is x^+i/- than 2xy ? 22. If d represents the diameter of a circle and c the circumference, we know that c=S}d. Find c when d=14. Find d when c = 22. 23. If ^ represents the area of a circle and r the radius, A = 3]r^. Find A when r— 7 ; when r=14. 24. By arranging the factors in the most suitable order, find th( values of 2*. 53. 252.43, 125.25. 8. Terms of an Expression. The parts of an algebraic expression which are connectec' by the signs of addition or subtraction are called the terms of the exuression ALGEBRAIC NOTATION 9 Thus, the expression 2a+36 has two terms, and the expression ix* — 3xy—y* has three terms. Quantities which are connected by the signs of multipUcation or division are not different terms. a* Thus, 4oa; is only one term, so is -r- 9. CoefiBcient. In the product 4a;, 4 is called the coefficient, or co-factor, of x. In ah, a is the coefficient of b and b is the coefficient of a. The 4 is a numerical coefficient, and the a or 6 is a literal coefficient. In any product, any factor is called the coefficient of the rest of the product. Thus, in 5abx, 5 is the coefficient of abx, 5a is the coefficient of bx, and 5ax is the coefficient of b. In any term where the numerical coefficient is not stated, the coefficient 1 is understood. Thus, in xy the numerical coefficient is L 10. Addition and Subtraction of Like Terms. When terms do not differ or differ only in their numerical coefficients, they are called like terms. Thus, 2ab, 5ab, ^ab are like terms, but 3a, 46, Gab are unlike terms. In arithmetic, quantities which have the same denominations may be added or subtracted. Thus, 3ft. -f 4ft. - 2ft. = 5ft. $12-$10+$8-$3 = $7. We cannot add or subtract quantities of different denomina- tions, unless we can first reduce them to the same denomination. Similarly, in algebra, like terms may be added or subtracted. Thus, 5a+ 2a = 7a, 5a — 2a = 3o, 6a6-f 5a6- 3a6= 1 lo6- 3o6 = 8a6, Sx*-6x*+9x*-2x*=\lx'~Sx'=--9x*. In the laat example we may, of course, perform the operations in the •^der in which they occur and obtain the same result 10 ALGEBRA Unlike terms can not be added or subtracted. Thus, the sum of 3a and 56 can be indicated in the form 3a4-56, but they can not be combined into a single term vmless the numerical values of a and 6 are given. BXBRCISE 4 (1-8, Oral) 1. What is the numerical coefficient of each term in the expression 5a3+a2+|o ? 2. What is the sum of the numerical coefficients in 2x^ + 3xy+x+y ? 3. Which are like terms in the expression 5a24-26-3a+76-4a2 ? 4. In 66cy, what is the coefficient of bey 1 oi cy 1 oi by 1 of 6 ? 5. What is the sum of : (1) 2a, 3a, 4a. (2) 5m, ^m, fm. (3) 4a2, la\ 5a^. (4) 3xy, 4xy, 2xy. 6. If x=2, find the numerical value of the sum of Sa;^ and 4x^ in two different ways and compare the results. 7. Simplify the expression 3a+86+2a+6+a+3fc by combining like terms. 8. Express in as simple a form as possible : (1) 5TO4-7m— 3m— 2m. (2) %ah—3ab + 2ab—ab. (3) 3x+a+2x+a. (4) 15a+106-7a-(-46. 9.* Combine the like terms in the expression : 2x+7t/+5z-a:+2i/-3z+3a;-4t/-z and find its value when a;=3, i/=5, z=10. 10. If a =6, find the value of 15o2— lOa^— 3a2+8a-5a— 20. 11. What arithmetical number does lOOa+106+c represent when a=2, 6=3, c=4 ? When a=9, 6=5, c=7 ? 12. SimpUfy 2x^-\-3x-\-l-x'^-{-\\x-2-x^-4x+b. 13. A man walks 4x feet East, then x feet West, then 3a; feet East, then 5x feet West. How far is he now from the starting point and in what direction from it ? ALOEBRAIC NOTATION 11 14. A man began to work for a firm on a salary of x dollars a year. If his salary for each year was double the salary for the preceding year, how much did he earn in four years ? 15. If x-\-Sz-\-5x is equal to 72, what is the value of x ? How do you know that your answer is correct ? 16. Write in the shortest form you can aaa+aaa-{-aa-\-aa-\-a-\-a-\-a. 17. Find the average of (1) 10, 8, 15, (2) 3x, Ix, 5x. 11. Use of Brackets. In algebra brackets are used for the same purpose and with the same meanings as in arithmetic. In finding the value of 10+8+5, we may perform the additions in any order, but if we wi'ite it 10+ (8+5), it is understood that the 8 and 5 are first to be added and the sum of 10 and the result is to be taken. Similarly, a-\-{b-\-c) means that the sum of the numbers represented by b and c is to be added to the number represented by a. In the expression 7+5x2, the multipUcation is to be performed first, and then the addition. If, however, we wish the value of (7+5) x 2, we must add the 7 and 5 before multiplying by 2. Although 10+(8+5) is equal to 10+8+5, it is clear that (7+5) X 2 is not equal to 7+5x2, the former being equal to 24 and the latter 17. When a is to be multiplied by b, the sign of multiphcation is omitted m the indicated product ; so when (7+5) is to be multiphed by 2 we may write 2(7+5) or (7+5)2, the sign of multiphcation being understood. It is thus seen that one of the uses of brackets is to indicate the order in which operations are to be performed. Thus, 10 — (7 — 3) means that 3 is to be subtracted from 7 and the result is to be subtracted from 10. If the values of the letters were given, what operations would you perform to find the values of : a+{b+c), a-{b+:), a-{b-c), {a-b)-{c-d)t The pupil should recognize that 3a^ is not the same as 12 ALGEBRA (3a) 2. The latter means that a is first to be multiplied by 3 and the product is to be squared. Thu8,ifa=2, 3a* = 3x4=12, and (3a)* = 3ax3a = 6x6 = 36. Brackets also indicate thai the numbers within the brackets are to be considered as a single quantity, that is,_ they are used for the purpose of grouping. The dividing line between the numerator and denominator of a fraction has the same value as a pair of brackets. Thus, in -, a-\-b is a single quantity and so is c + d. The fractional c-\-d form is another way of writing (o+6)-7-(c+d). E3XB3RCISB 5 (1-18, Oral) Perform the operations indicated : 1. 10-(6+3). 2. 8-(4-2). 3. 15-6-^3. 4. (15-6)-i-3. 5. 3(4+7-5). 6. (10+2)(5-l). 7. 10+2x5-1. 8. (16+12)^(6-2). 9. lx-{8x-4z). 10. (6a-2a)-(7a-4a), 11. (3a;+4a;)^7. 12. (10a-6a)^2a. 13. (3a;+9a;)-f-(6a;— 3x). 14, 3(7-5)-2(8-6). 16. 43x— (7a;— 4a;)+2x. 16. x-{^y+3y-ly). 17. (56-46)(32-2z). 18. 6a— (la— 3a) 8a-(6a+a) Indicate, using brackets : 19. That X is to be added to the sum of p and q. 20. That the sum of x and y is to be added to m. 21. That the sum of a and 6 is to be multiplied by 2. 22. That the difference of m and n, where m is greater than n, is to be subtracted from a. 23. If p is greater than q, that the difference of p and j is to be divided by the sum of m and n. If a=lO, 6=3, c=2, find the value of : 24.* Sa-(2b-^c)-5(a~bV 26. 7(o-6-c)-3(a-26+c). ALGEBRAIC NOTATION 13 26. (3o+26— c)(o— 36). 27. a^+b^+c^—2{ab-\-bc-\'Ca). a-\-Zh—c 2a— 36— 3 a— 6 b—c c+a 2a—5b+2c~ a+b—2c ' ' be '^ c+a o+26 " 30. When a=6 and 6=3, show that R(o— 6) +3(a+6) = 2(4a-6). EXERCISE 6 (Review of Chapter I) 1. If a; represents a certain number, what does 4a; represent ? ^xl X* 1 3x* ? 2. If a represents a number, what will represent 5 times the number ? y times the number ? 3. How do you indicate that y is to be added to x ? That x is to be subtracted from y ? 4. Indicate the sum of x and y diminished by a. 5. If one yard of cloth costs x cents, how many cents will 10 yards cost ? How many dollars ? 6. If a yard of ribbon is worth y cents, how much is a foot worth ? 7. A man bought an article for x dollars and sold it at a loss of y dollars. What did he sell it for ? 8. If I paid a dollars for b articles, how much did I pay for each ? What would c articles cost at the same price ? 9. A boy has a dollars. He earns b cents and then spends c cents. How many cents has he left ? 10. I have x dollars. If I pay two debts of a dollars and b dollars, how much shall I have left ? 11. If one number is x and another is 5 times as large, what is the sum of the numbers ? 12. If one part of 10 is x, what is the other part 7 13. A man worked rn hours a day for 6 days. If he was paid $2 per hour, how much did he earn ? 14. How far can a man walk in 5 hours at 4 miles per hour 7 In a hours at b miles per hour ? 15. A man bought x acres of land at a dollars per acre and sold it at a loss of 6 dollars per acre. What did he sell it for ? 16. What number is 15 greater than x ? 15 less than x 7 17. By how much does a* exceed 6* when a = 7, 6 = 3 T i4 ALGEBRA 18. When x=l, j/»=2, z = 3, what are the values of x-\-y-"t, 2x+5y-3z, 7x-3y+z T 19. If a = 3, 6 = 4, c=0, find the values of 2ab, Aac, a'+b*+c>, 6a»-26*+4c». 20. If x=^ and y=s, what are the values of 3x — 2y, 6xy, 2x* — 3y*, 8x»- 272/3 ? 21. If a= 10, 6 = 5, c = 3, find the values of a(6+c), a(6— c), a{6* — c*), c(a»-6»). 22. What is the sum of 2x, 5x, Ix and 3x ? 23. Simplify 5a— 3a+lla+o-10o. 24. What is the average of 20, 15, 0, 8, 12 ? Of 2a, 3a, la ? Ot a, 6, c, d ? 25. In 8 years a man will be x years old. How old was he 8 years ago ? 26. B has $20 more than A, C has $20 more than B. If A has $«, how much has C 7 27. What is the sum of the numerical coefficients in the expression 3a+ ^a6+ac+ fad ? 28. Express 1000 aa a power of 10 ; 32 as a power of 2 ; 81 as a power of 3 ; 64 as a power of 4. 29. Express 15, 105, 3a6, 35x*y, as the products of simple factors. 30.* How long will it take me to walk o miles at 3 miles per houi and ride 6 miles at 12 miles per hour ? 31. A farmer buys 5 lb. of tea at x cents per lb. and 20 lb. of sugar at y cents per lb. He gives in exchange 7 lb. of butter at z cents per lb. If he still owes something, how much is it ? 32. If I buy 100 lb. of nails at a cents per lb. and 200 lb. at 6 centf per lb., what is the average cost per lb. ? 33. What is the total number of cents in x five-cent pieces, y ten- cent pieces and z half-dollars ? 34. What number is represented by lOOOa+1006-f lOc + d, when a=I, 6=2, c = 3, d=4 ? When a=4, 6 = 0, c=l, d = 9 ? 35. When a = -2 and 6 = -I. what are the values of a+6, ab, ti ^, a»+6», a»-6» 7 ab ALGEBRAIC NOTATION 16 36. li A can do a piece of work in 10 days and S in 15 days, what fraction of the work can they together do in 1 day ? What fraction if A could do it in x days and B in y days ? 37. If a = 20, 6=15, c=10, d = 5, find the difference between (0+6) — (c+d) and (o — 6) — (c — d), also between 3(a+6) — 5(c — d) and 6(0 -(f)- 3(6 -c). 38. When x = 7 and y=\, the product of x-\-y, x+2y, x — by is how much greater than the product of x — y, x—2y, x—3y ? 39. If = 3, find the value of 1 -\ ^ • CHAPTER II SIMPLE EQUATIONS 12. Idea of Equality. In weighing an article, when you see that the scales are balanced, what conclusion do you draw ? If a 5 lb. bag of salt is placed in one scale pan, what weight (w) must be placed in the other pan to restore the balance ? What must w be to balance a 3 lb. bag and a 4 lb. bag ? If the scales are balanced in each of the following figures, what must w be equal to ? Fia. 1. Pio. 2. Fio. 3. If U)+4 = 9, as in fig. 1, what is w equal to ? If w-\-io= 12, as in fig. 2, what is iv equal to ? If 1^+3 = 5+2, as in fig. 3, what is w equal to ? If the scales are balanced and I add 2 lb. to one side, what else must I do to preserve the balance ? What, if I take away 3 lb. from one side ? If I double the weights on one side ? If I halve the weights on one side ? 13. The Equation. When a certain number is added to 10 the result is 27. What is the number ? The condition expressed in this problem might be more briefly shown in the form : 10+ a certain number =27, or in the form 10+? =27, where the question mark stands for the required number. Any other symbol would answer the same purpose as the SIMPLE EQUATIONS 17 question mark. Thus, if x represents the required number, then the problem states that 10+a;=27. This statement is called an equation and is merely a short way of stating what is given in the arithmetical problem preceding. In order that the statement may be true, it is easily seen that the symbol x must stand for the number 17. Ex. — When ? number is multiphed by 3, and 5 is sub- tracted from the product, the result is 19. What is the number ? Here, if x stands for the number, the problem states that 3a:-5=19. Before the 5 was subtracted the product was evidently 5 more than 19 or 19+5 or 24. If 3 times the number is 24, then the number must be \ of 24 or 8. The solution may be written more briefly thus : If 3a;-5=19, .-. 3a;= 19+5 = 24, x=Jof24 = 8. That 8 is the correct value for the number is shown by the fact, that when it is multiplied by 3 and 5 is subtracted from the product, the result is 19. 14. Solving an Equation. The process of finding the value of X, such that 8a::— 5=19, is called "solving the equation," and the value found for x is called the root of the equation. ESXSRCISS 7 (Oral) 1. State the number for which the question mark stands in each of the following : (1) 5+? = 12. (2) ? + 12=20. (3) 10-V=2. (4) 15=-8 + ?. (5) 40=62-?. (6) ?-8=42. 2. What is the number for which x stands in each of the following : (1) a:+6=20. (2) 8+a;=32. (3) 25=x+6. (4) a:-16=7. (5) I0-a:=8. (6) 12=17-x. O IB ALGEBRA 3. The first liquation in Ex. 2 states that when a number is in creased by 6 the result is 20. What does each of the other equa^Bone say ? 4. If 3 times a number is 45, what is the number ? If one-hali of a number is 16, what is the number ? If n stands for a given numberj what would represent \ of the number ? \ of the number ? 5. For what number does n stand in each of the following equations* (1) 4«=24. (2) ^n = 10. (3) |m=36. (4) ln=U. 6. If 2x+5=ll, what is the value of 2x ? oi xl 7. If 3/rt— 2=13, what is the value of 3m ? of m T 8. If ^p+3=10, what is the value of |p ? of p ? 9. If -^x— 11=7, what is the value of ;ix ? of a; ? 10. If 2(a;+4) = 14, what is the value of a:+4 ? of a; 7 Solve the equations : 11. a;+10=30. 12. 3x— 2=16. 13. 5)/ + 2---17. 14. 4<-5=27. 15, 2n = ll. 16. 7?i-4=24. 17. 3m;+2 = 38. 18. ^x— 1=4. 19. 2/i+l=4. 20. 3w— J=5J. 21. iM;+2=5. 22. |x-5=15. 23. 3(x+l) = 30. 24. 5(x-2)=45. 25. i(x-l)=3. 15. Axioms used in Solving Equations. If two numbers are equal, what is the result when the same number is added to each ? Thus, if x=6, what is x+2 equal to T What is the result when the same number is subtracted from two equal numbers ; or when each is multiplied by the same number ; or when each is divided by the same number ? Thus, if x= 10, what is x— 4 equal to 7 What is 3x equal to ? What ia \x equal to ? The preceding conclusions may be stated thus : (1) // the same number he added to equal numbers, the sums are equal. (2) // the same number he subtracted from equal numbers^ (he remainders art equal. SIMPLE EQUATIONS 19 (3) // equal numbers be multiplied by the same number, the products are equal. (4) // equal numbers be divided by the same number, the quotients are equal. These statements are called axioms, or self-evident truths, and are used in solving equations. The method is illustrated by the following examples : Ex. 1.— Solve 3a:-7=35. Add 7 to each side, .-. 3x— 7+7 = 35 + 7, axiom (1), .-. 3x = 42. Divide each side by 3, .". x= V = l'i> axiom (4). Ex. 2.— Solve ia;+2=34. Subtract 2 from each side, .-. ix+ 2-2 = 34-2, axiom (2), ^x=32. •Multiply each side by 2, .". x=64, axiom (3). Ex. 3.— Solve 5x— 3=2a;+12. Add 3 to each side, .'. 5x=2a;+15. Subtract 2x from each side, .". 5x— 2x=15, 3x=15. Divide each side by 3, .'. x = 5. The object of the changes which have been made in these equations is to get the quantities containing the unknown {x) to one side and the remaining quantities to the other side. The unknown quantities are usually' transferred to the left side, but sometimes it is better to transfer them to the right. Ex. 4.— Solve 3m+20=5m— 16. Add 16 to each side, .*. 3m+36=5m, Subtract 3m from each side, .'. 3m+36 — 3m = 5m — 3m, 36 = 2m, 18 = m or m= 18. 16. Verifying the Result. If we substitute 18 for m in the first side of the last equation we get 3m+20=3x 18 + 20=74. If we substitute in the second side we get 5m-16=5x 18-16=74. 02 20 ALGEBRA This process is called verifying or testing the correctness of the result. If the root obtained is the correct one, the two sides of the equation should be equal to the same number when the value found for the unkno\^Ti is substituted. The equation is then said to be satisfled. The beginner is advised to verify the result in every case. Verify the results obtained in Ex.'s 1, 2 and 3. BXBRCISB 8 (Oral) i. If 3x=lo, what does x equal ? What axiom is used ? 2. If oar -1-2 = 17, what does 5^ equal ? What axiom is used ? What does x equal ? What axiom Ls used ? 3. If 2?/— 3 = 13, what does ?/ equal ? What two axioms are used ? 4. If Ax— 4=6, what does ^x equal ? What does x equal ? What two axioms are used ? 5. If ?x=6, what does \x equal ? What does x equal ? What two axioms are used ? What is the value of x in the following equations : 6. 2x=18. 7. 6x=72. 8. 5x=:16. 9. 3x=6-9. 10. x+20=25. 11. 2x4-1 = 15. 12. 3x-l=20. 13. 6x+5=29. 14. ix=8 15. §x=12. 16. Jx=2|. 17. ^x=15. BXHIRCISB 9 Solve the following equations, giving full statements of the methods In each case verify the result : 1. 3x-rll=47. 2. 2x+5=27. 4. 3x- 10=65. 5. 4x=x+21. 7. 7x=60+3x. 8. ix+5=50. 10. 10x-L3=3x-f 66. 11. 6a— 3a=a+5. 13. 8m=36— 4m. 14. 20+6x-|-5=50-3x-fll. 15. 12x-652=7x+428. 16. 764x-9=680x-f 12. 17. Nine blocks of equal weights (w) together with a 20-gram weight are balanced by weights of 50 grams and 10 grams. Express this by an equation and find the weight of each block. 18. If 17x— 11 is equal in value to 5x-t-121, what is the value of x ? 19. What value of j/ will make 11?/ +60 equal to 20t/— 30 7 3. 4x-5=51. 6. 4y=2y+80. 9. 6x+42=9x. 2. 10x+20=20. SIMPLE EQUATIONS 21 17. As we have already shown, an equation is merely the statemeni in algebraic form of the condition given in an arithmetical problem. The solution of the problem is thus obtained by solving the equation. EXBRCISS 10 State the condition in each of the following problems in the form af an equation : 1. What must be added to 33 to make 50 ? 2. What must be taken from 90 to leave 40 ? 3. What is the number which when doubled is 36 ? 4. Five times a certain number is 45. What is the number ? 5. If a number is doubled and 3 added, the result is 25. What is the number ? 6. What number is doubled by adding 27 ? 7. What number is halved by subtracting 20 ? 8. If 8 is subtracted from f of a certain number, the result is 7 What is the number ? 9. Solve the equation in each of the preceding examples. 18. Problems Solved by Equations. The following examples will illustrate the method of solving problems by means of equations : Ex. 1. — When I double a certain number and add 16, the result is 40. What is the number ? Let X represent the required number. Then 2x»tS the double of the number. Then 2x+16 is the double with 16 added. ' But the problem states that this is 40, .-. 2a;+ 16 = 40, 2x=24, a;=12. Therefore the required number is 12. The result should be verified by showing that the number obtained satisfies the given problem. Verification; When 12 is doubled I get 24 eind when 16 ia added I got 40= Therefore the result is correct 22 ALGEBRA Note that the substitution is made in the original, problem, not in the equation. There might be an error in writing down the equation and then the sohition obtained might satisfy the equation, but would not necessarily satisfy the given problem. Ex. 2. — The number of pupils in a class is 33, and the number of boys is 7 greater than the number of girls. Find the number of each. Let x = the number of girls, x+7 = the number of boys, .'. x-\-x-{-l = the total number, .-. x+a;+7 = 33. 2a; = 33-7 = 26, a;=I3, .-. x+l = 20, .'. the number of girls is 13 and the number of boys is 20. Verification : 20+13 = 33,20-13 = 7. Ex. 3. — Divide §100 among A, B and C, so that B may receive 3 times as much as A, and C S30 more than B. Let x= the number of dollars A receives, 3x= „ „ „ B „ :. 3x+30= „ „ „ c .'. they all receive (x+3a;+3x+30) dollars, .-. a;+3x+3x+30=100, 7x+30=100, 7x = 70, a;=10, .-. A receives $10, B $30 and C $60. Verify this result. 19. Steps in the Solution of a Problem. The examples which have been given will show that in solving a problem the steps in the work are usually in the following order : (1) Read the problem carefully to see what quantity is to be found. (2) Represent this unknmvn by a letter. (3) If there be more than one quantity to be found, represent the others in terms of the same letter. SIMPLE EQUATIONS 23 (4) Express the condition stated in the problem in the form of an equation. (5) Solve the equation and draw the conclusion. (6) Verify the solution by substitution in the problem. On referring to Ex. 1, we see that there was only one quantity to be found, and therefore step (3) did not appear in the solution. In Ex. 2 there were two quantities to be found, and when we represented the number of girls by z, we could represent the number of boys by x+1. The pupil is advised to make full statements, in plain English, as to what the unknown represents. Thus, in Ex. 3 to say, let x=A, or let x=^A's money, will only lead to difficulties. Note. — The examples in the following exercise are to be solved by means of the equation and the results should be verified in every case. Although the answers to many of them may be given mentally, the pupil is advised to give complete solutions, so that he may become familiar with algebraic methods. E3XERCISE! 11 1. If 37 is added to a certain number, the sum is 63. What is the number ? 2. If 27 is subtracted from a number, the result is 5. What is the number ? 3. A number was doubled and the result was increased by 27. If the sum is now 73, what was the number ? 4. When a number is multiplied by 7, and 25 subtracted from the product, the result is 59. Find the number. 5. If five times a number be increased by 6, the sum is the same as if twice the number were increased by 15. Find the number. 6. What number if trebled and the result diminished by 36 gives twice the original number ? 7. If you add 19 to a certain number the sum is the same aa if you add 7 to twice the number. Find the number. 8. Five times a number, plus 19, equals nine times the number, minus 41. What is the numbw ? 24 ALGEBRA 9, Two numbers differ by 11 and their sum ia 51. Find the numbers. 10. The sum of two numbers is 47 and one exceeds the other by 15. What are the numbers ? 11. ^'s salary is three times B's and the difference of their salaries is S1500. Find the salary of each. 12. A horse and carriage are worth $360. The carriage is worth twice as much as the horse. Find the value of each. 13. Divide 93 into two parts so that one part will be 27 less than the other. 14. The length of a rectangle is three times the width. The peri- meter is 72 feet. Find the sides. 15. ^ is twice as old as B. In 10 years the sum of their ages wUl be 41 years. What are their ages ? 16. Divide $500 between A and B so that A will receive $20 more than twice what B wUl receive. 17. The sum of two consecutive numbers is 59. What are the numbers? (Let x be the smaller number, then x+1 will be the greater. ) 18. Find three consecutive numbers whose sum is 150. 19. ^'s age is twice B's and C is 7 years older than A. The sum of their ages is 67 years. Find the age of each. 20. The difference between the length and width of a rectangle is 10 feet and the perimeter is 68 feet. Find the sides. 21. Divide S468 among A, B and C, so that B may get twice as much as A, and C three times as much as B. 22 . A railwav train travels m miles per hour. If it goes from Toronto to Montreal, a distance of 333 miles, in 9 hours 15 minutes, what is the value of m ? 2.3. A line 20 inches long is divided into two parts. The length of the longer part is ^ inch more than double the shorter one. Find the lengths of the parts. 24. What value of x will make 5x-|-6 equal to 3x -f 40 ? 25. If 5"o of a sum is $48, what is the sum ? 26. An article sold for $2-61, the loss being 10%. What was the oost 7 SIMPLE EQUATIONS 26 27. Divide $1496 among A, B and C, so that B will get three times J's share and C wili get $100 more than A and B together. 28. A has five times as much money as B. After A has spent $63 he has only twice as much as B. How much has B ? 29. If $20 less than | of a sum of money is $10 more than ^ of it what is the sum ? 30. Three boys sold 42 papers. The first sold J as many as the third and the second sold \ as many as the third. Kow many did each sell ? 31. The sum of ^ of a number and \ of the same number is 55. What is the number ? 32. A man paid a debt of $4500 in 4 months, paying each month twice as much as the month before. How much did he pay the hrst month ? 33. The half, third and fourth parts of a certain number together make 52. Find the number. 34. Divide 72 into three parts so that the first part is J of the second and J of the third. 35. What number is that to which if you add its half and take away its third, the remainder will be 98 ? 36. n 3a = 46c, . (1) Find a, when 6:= 10, c= 15. (2) Find b, when o =12, c= 3. (3) Find c, when o= 8. 6= |. BXBROISE) 12 (Review of Chapter II) 1. State the four axioms which are used in solving equations. 2. Show that x=18 is the correct solution of the equation 3a;— 7 = 2a;+ll. 3. Determine if 8 is a root of 3(a;+6) = 5(a;— 1). 4. Solve (a) 5x4-3 = 2x+ 9; (6) l+2x=9-2x ; (c) 3x--7 = 8-2x ; (d) 7x+l = 9a;-9; (e) \\x—\ = 5x+\. 5. My house and lot cost $16,800, the house costing five times as much as the lot. Find the cost of each. 6. A horse and carriage cost $520. If the carriage cost $60 more fehsm the horse, what did the horse cost 5^ 26 ALUEBKA 7. Three farmera together raised 2700 bushels of wheat. A raised three times as much as B, and C raised twice as much as A. How much did each raise ? 8. What value of x will make 136 — 3a; equal to 172 — 9x ? 9. Where r is the radius of a circle and c is the circumference, c=2irr, where Tr = 3y. (a) Find c, when r= 7 ; when r=42. - (6) Find r, when c = 88 ; when c=ll. 10. If 5=J/<«, find 5 when « = 4 and/=32; when <= 10 and /= 32-2, 11. In a company of 98 persons, there are twice as many women as men, and twice as many children as women. How many children are there 7 12. Six boys and 15 men earn 5264 a week. If each man earns four times as much as each boy, how much does a boy earn in a week T 13. Five times a certain number, increased by 47 is equal to eight times the number, diminished by 43. What is the number ? 14. An agent charges 3 % commission for collecting an account. If his charge is $11-13, what was the amount of the account ? 16. Solve (a) •05x=4; (6) x+-04x=208 ; (c) x-06x=235 ; \d) a;+6%x=630. 16. If 6 is the base of a triangle and h is its height, the area (o) is given by the formula a=\bh. (i) Find a, when b= 8, /t= 4. (ii) Find 6, when a = 36,71=12. (iii) Find h, when a=176, 6=22. 17 . The sum of the unequal sides of a rectangle is 65 feet and their difference is 15 feet. Find the area of the rectangle. 18. If 6x — i/ = 2x + 2/> what is the value of y if x = 6 ? 19. For what number does the question mark stand, if 5x+i = 3x+? is satisfied when x = 3 7 20. If 4 % of X together with 3 % of x is equal to 35, find x. 21. State a problem the condition of which is expressed by the equation 3x— 20=x. 22. B has $10 more than A, and C has $20 more than B. Together they have $190 How much has each 7 23. A turkey costs as much as three chickens. If 2 turkeys and 3 chickena cost $7-20, find the cost of a chicken. SIMPLE EQUATIONS 27 24. What number increased by J of itself is equal to 60 ? 25. Divide $6400 among A, B and G, so that B will get $120 more than A, and C $160 more than A. 26. The net income from an enterprise doubled each year for five years. If the total net income for the five years was $7750, what was the income for the first year ? 27. If2a6 = 3mn, (1) Find a, when 6=15, m= 6, n= 5. (2) Find b, when a= 12, w= 2, n= 2. (3) Find m, when a= i^, 6 = 6, n = J. (4) Find n, when o= -3, 6 =-6, m = -12. 28. Show that 6 is a root of the equation 2(x-l)(a;+2) = 4(x+3)(a;-5) + (x-2)(a;+5). 29. The area of the United States is 4000 square miles more than seventy times the area of England. If the area of the United States is 3,560,000 square miles, find the area of England. 30 Solve and verify : (1) 6850+a; = 27a;+350. (2) Jx+Jx+}x = 3380. (3) 16O7x+20=1762z-ll. CHAPTER in POSITIVE AND NEGATIVE NUMBERS 62 51 60 59 58 57 66 65 54 53 62 \ \ / \ i \ 1 \ 1 \, J \ / \ / \ / 20. Arithmetical Numbers. In the diagram the hours from 12 noon to 12 midnight are represented on the horizontal line, and the temperature at each hour is shown by the position of a point on the corresponding vertical line. Thus at 3 P.M. the tempera- ture was 61°, at 7 P.M. 53° and at 11 P.M. .55-5°. The points which show the tem- perature for each hour are con- nected by a curve. This curve gives a picture of the changes in temperature during these twelve hours. These changes might be shown by a column of figures, but the curve exhibits the variations in temperature more readilj' to the e\'e. We can see at a glance when the tempera- ture was rising and when faUing, at what hours it was the same, that it rose or fell more rapidly during certain hours than during others. Here we say that we have represented graphically the changes in temperature, and the curve shown is called a graph. 121 2 3 4 5 6 7 8 9 1011 12 ■XERCISB 13 (1-8, Oral) Using the diagram, answer questions 1-8. 1. What was the temperature at 1 p.m., at 4 p.m., at 10 P.M. T 2. At what hours wm the temperature the same 7 38 POSITIVE AND NEGATIVE NUMBERS 29 3. What was the highest temperature ? What the lowest ? 4. What was the range of temperature ? 6. Between what hours was it rising ? 6. How much did it rise between 10 and 11 ? How much did it fall between 6 and 7 ? 7. When was it 60°, 58°, 55° ? 8. Between what hours did it rise most rapidly ? When did it fall most rapidly ? 9. The percentage of games won by a baseball team, up to the beginning of each month of the playing season, was as follows : June, 66 ; July, 63 ; Aug., 60-5 ; Sept., 62 ; Oct., 61-5. Draw a graph showing these changes. 10. A boy's height in inches, for each year from the age of 7 to the age of 14, was 44, 47, 50, 51, 52-5, 54, 56-5, 58. Draw a graph to illustrate the variations in his height. 21. Negative Numbers. This diagram shows the average temperature for a week. Thus on Monday it was 25° above zero, while on Thursday it was 15° below zero. We might express this algebra- ically by saying that on Monday the temperature was +25°, and on Thursday it was — 15°. The number +25 is called a positive number and is read " posi- tive " 25 or " plus " 25, while —25 is called a negative number, and is read " negative " 25 or " minus " 25. A negative number is therefore one which is measured on the opposite side of zero from a positive number. sxsrcise: 14 1. Using algebraic signs, write down the temperature for each day in the diagram. Also read the temperature. ' 2. On what days was the temperature negative ? 30 ALOE BR A 3. How much higher was it on Monday than on Thursday ? How much lower on Tuesday than on Saturday ? 4. If the temperature is —30° and it rises 40°, how much will it be then ? If it had fallen 10", how much Avould it have been then ? 5. The temperature at which mercury freezes is — 39°C. What does that mean ? How much lower is it than the normal temperature of the blood which is 4-37°C. ? 6. If the price at which a certain stock sells above par is positive and the price below par is negative, make a diagram similar to the preceding, showing the prices of a certain stock for a week, when the record was as follows : Mon., 4 above par ; Tues., 2 below ; Wed., 1 above, Thurs., at par; Fri., 3 below; Sat., H below. 22. Distances measured on a Horizontal Line. 9 B O AC I \ 1 \ 1 + \ \ \ 1 1 -5 -4 -3 -2 -1 *l +2 ^3 +4 +5 On this diagram the distance between each successive marking represents one foot. What is the length of OA ? of OB ? In what respect does OA differ from OB 1 How might we use signs to show this difference ? It is usual to consider measurements made to the right as positive and to the left as negative. What point is +5 feet from ? What one is —5 feet from T If a point moves from O, 4 feet to the right and then 7 feet to the left, how far is it then from O ? Is the distance positive or negative ? We thus see that in addition to the numbers of arithmetic which begin with zero and extend indefinitely in one direction, we now have another series of numbers which also begin with zero and extend indefinitely in the opposite direction. In each series all integral and fractional numbers are included. 23. Further Example^ of Negative Numbers. (1) A man has property worth $100, and debts amounting POSITIVE AND NEGATIVE NUMBERS 31 to $60. When he has paid his debts he wiU have property worth $40. Thus, $100- $60= $40. If, however, he has debts amounting to $100, when these are paid he will have nothing left. Thus, $100- $100= $0. If he has debts amounting to $140, when he has paid a'.' Ae can he will still owe $4G. We express this algebraicall thus : $100- $140= - $40. In the first case we say that his net assets are $40, in the second they are zero, and in the third they are miniis $40. When we say his assets are — $40, we mean he is ji40 in debt. It will be seen that the difference in mea'iing between +40 and —40 when referring to dollars is practically the same as the difference between +40 and — 40, when referring to degrees of temperature, as in art. 21, or to distances measured in opposite directions on a horizontal Line, as in art. 22. (2) If a man gains $20 on one transaction and loses $15 on another, what is the net result ? If he had lost $25 on the second transaction what would have been the net result ? If we attach a plus sign to the result when it is a gain, how may we indicate a loss ? If O represents a sum gained ^nd L a sum lost, state the result in each of the following, attaching tt proper sign : 1. $30G+$20(?. 2. $30 04 $20 i. 3. $30Z,+ S20i. 4. $30 L+ $20 O. 5. $40 O^ $4f L. 6. $20 6+ $60 L. (3) If a game won is represented by +1, then —1 would represent a game lost. In a series of games I find that my record is : won, lost, lost, won, lost, won, lost, won, won. This might be represented thus : + 1-1-1 + l-l + l-l + l + l = +5-4= + l. What does this result mean ? Write in a similar way the following record : lost, lost, won, lost, won, lost, lost, won. Also the following : won, lost, drawn, won, won. 32 ALGEBRA (4) In locating points on the earth's surface, the distance in degrees north of the equator (north latitude) is said to be positive, and south of the equator negative. Thiis, the latitude of Toronto is + 44° and of Rio de Janeiro is - 23°. What is the distance in degrees of latitude between these two cities ? The preceding illustrations show that a positive number differs from a negative number in direction or quality. Thus, if +10 means 10 yards measured to the right; or 10° east longitude ; or 10 games won ; or 10 miles a boat goes up stream ; or 10 minutes a clock is fast ; or $10 in my bank balance ; or 10 pounds lifted by a balloon ; what would — 10 mean in the corres- ponding cases ? 24. Signs of Operation and Sighs of Quality. The numbers +25 and —25 are the same in magnitude, but differ in direction or quality. When a number is preceded by the sign +, it means that the number is taken in the positive direction or sense, and when preceded by the sign — , that it is taken in the negative direction. It will thus be seen that we use the signs + and — with two different significations. When they are used to indicate the operations of addition or subtraction, they are called signs of operation. When they are used to indicate direction or quality, they are sometimes called signs of quality. The beginner might think that this ambiguity would lead to confusion, but he will find that such is not the case. When we read a quantitj'' hke —25, we should say " negative 25," but this is not followed in practice, as it is usually read " minus 25." When no sign precedes a number, it is understood to be a positive number. 25. Absolute Value. The absolute value of a number is its value without regard to sign. Thus, 4- S and — 8 have the same absolute valu'' POSITIVE AND NEGATIVE NUMBERS 33 BXEBCISB 15 (1-15, Oral) 1. What is the net property of a man who, (a) has $60 and owes $47, (6) has $40 and owes $50, (c) has $65 and owes $65 ? 2. What is the value of, (a) $40- $30, (6) $40— $60, (c) $30— $20, (d) $20- $30, (e) $10- $0, (/) $0-$10? 3. The temperature was —10° at 6 p.m. and 4° at 10 p.m. How many degrees did it rise in the interval ? 4. A Uquid whose temperature is 20° is cooled through 30°. What is the final temperature ? 6. A vessel sailed on a meridian from latitude 15° to latitude —5°. How many degrees did it sail and in what direction ? 6. What is the distance between a place 90 miles due east of Toronto and another 60 miles due west ? 7. I am overdrawn at the bank $20. What must I deposit to make my balance $100 ? If — 20-|-x=100, what is x 1 8. What would a negative number mean in stating the height of a tree above the window of a house ? The height above sea level of the bottom of a well ? 9. A man biiys a horse for $100 and sells him for $80. What is his gain and his gain % ? 10. A man travels 20 miles from A and his friend travels — 10 mile* from A. How far are they apart ? 11. What is the rise in temperature from —30° to —10° ? If — 30-t-a;= — 10, what is x t 12. What is the distance between two places which are a Miles and b miles west of Montreal, (1) if a is greater than b, (2) if b is greater than a ? 13. Denoting a date a.d. by + and B.C. by — , state the number of years between these pairs of dates : (1) +1815 to +1915. (2) -20 to +75. (3) -65 to -37. (4) -120 to +60. (5) -200 to +200. (6) +1900 to +1800 14. Augustus was Roman Emperor from —31 to +14. How many years was he Emperor 1 What is the difference betweei 14 and — 31 ? D 34 ALGEBRA 16, The First Punic War lasted from -264 to —241. How long did it last ? What is the difference between —241 and —264 ? 16. A boy adds 15 marbles to his supply, gives away 10, buys 5 and gives away 12. How many has he thus added to his supply ? 17. I have Sn in the bank. If I issue a cheque for S6, what is my balance when the cheque is paid ? If a=40 and 6=50, how do you mterpret the result ? 18. A has S50 and B has $20. A owes B $10 and B owes A $40. How much will each have when his debts are paid ? 19. The weights of two pieces of iron are 65 lb. and 147 lb. If they are attached to a balloon with an upward pull of 239 lb., how would you represent the combined weight ? 20. Represent graphically the following changes in the price of a stock : Month. July. Aug. Sep. Oct. Nov. Dec. Jan. Feb. Mar. Apr. Amou nt above par ... 6 1 4 5 2 3 Amou nt below par . . . 2 3 1 4 EXERCISE 16 (Review of Chapter III) 1. Using signs, express the results of the following transactions : (1) A gain of $10 followed by a loss of $15. (2) A loss of $12 followed by a loss of $4. (3) A loss of $8 followed by a gain of $10. 2. What is the difference between 40° and — 3° ? 3. If an upward force or pull is positive and a downward force is negative, what single force is equal in effect to these pairs of forces : (1) 10 1b., -3 1b. (2) 8 1b., -12 1b. (3) -7 lb., -2 lb. (4) -9 1b., 3 1b. (5) 6 1b., -6 1b. (6) 2olb., -alb.? 4. \n firinp at a target each hit counts 5 and each miss —3. If I fire 10 times and make 6 hits, what is my score ? If I make only 2 hits what is my score ? 5. What is the fall in temperature from 27° to — 1 1° 7 If 27-x= —11, what is X? POSITIVE AND NEGATIVE NUMBERS 35 6. In a 100 yards handicap race A has 3 yards start and B has — 3 yards start. What do these mean ? How far has each to run ? 7. In solving a problem in which it is required to find in how many years A will be twice as old as B, I get the answer — 10. What does this answer mean ? 8. Find the average noon temperature for a week in which the noon temperatures were : 20°, 10°, 15°, 0°, 4°, -6°, —15°. 9. A train was due at 10 minutes to 3. How many minutes before three did it arrive if it was half an hour late ? 10. A man travels 8 miles, then —6 miles, then 4 miles, then — 1 1 miles. How far has he travelled ? How far is he from the starting point and in what direction (positive or negative) from it ? 11. Egjrpt was a Roman province from —30 to 616. How many years was this ?. What is the difference between 616 and — 30 ? 12. The daily average temperature for 14 days were : 6°, 5°, 0°, -4°, 2°, -6°, -2°, 0°, 5°, 1°, -1°, -6°, -3°, 3°. Show these variations by means of a graph. 13. If a gain of a dollar be the positive unit, what will represent a loss of $3-50 ? 14. The record of a patient's temperature for each hour beginning at 12 noon was: 100°, 100-5°, 102°, 101°, 104°, 101-5°, 99-5°, 98°, 97-5°, 97°. Represent these changes graphically, taking two spaces on the vertical line to represent one degree. 15. If the normal temperature of the body is 98-5°, write the record in the preceding question using positive and negative signs. 16. The minimum temperatures for the first 15 days of December were: 26°, 22°, 14°, 25°, 21°, 18-5°, 13°, 7-5°, 11°, 6°, -4°. -6°, -1°, 10°. 12-5°. Make a chart to show these variations. d2 CHAPTER IV ADDITION AND SUBTRACTION 26. Addition of Positive Quantities. What is the result ol combining : (1) A gain of $20 with another gain of $10 ? (2) A measurement of 5 feet to the right with another of 3 feet to the right ? (3) A rise in temperature of 10° with a rise of 8° ? (4) 6 points won with 4 points won ? As explained in Chapter III., we will consider all of these to be positive quantities, and we might show this by attaching the positive sign to each. We might write these four questions as problems in addition, thus : + $20 +5 feet +10° + 6 points + $10 +3 feet +8° + 4 points + $30 +8 feet +18° +10 points Similarly, the sum of 6a and 4a is 10a, and the sum of 2x*, 5x* and 6x»i3 13x'. Here we have not prefixed any sign, and when that is the case the positive sign is understood. We see then that the sum of any number of positive quantities is always positive. 27. Addition of Negative Quantities. We might change the data of the four questions in the preceding article so that all the quantities would be negative. M ADDITION AND SUBTRACTION ST Thus, the first might be changed to — . "What is the result of combining a loss of $20 with a loss of $10 ? " Read the other three questions making similar changes. What would now be the answer to each question ? As problems in addition they would now appear thus : -$20 -5 feet -10° — 6 points — $10 -3 feet — 8° — 4 points — $30 - 8 feet — 18° - 10 points Similarly, the sum of — 7x and — 5x is — 12x, and the sum of —5a*, — 2o^, —a^ and —6a' is —14a'. Thus, the sum of any number of negative quantities is negative, and is found by adding their absolute values and prefixing the negative sign to the result. EXERCISE 17 (Oral) State the results of the following additions : 1. + $3 2. -$10 + $5 -$ 8 10. 12. 3. -12° -10° 4. 8. -3y, 3 yd. 5 yd. 5. - 7 6. 4a2 -13 6o2 Add 10a;, 12x, 15a;. Add Jm, f m, f m, |m. 7. — 10a;2j/ — 5x^y —^abc — ^abc 8. 11. . Add — 3m2, , Add -I2y, -ny. 28. Compound Expressions. An expression of one term is frequently called a simple expression, while one of more than one term is called a compound expression. i Thus, 2a, 3x^y, labc are simple expressions, and 2a + 36, 5x — 3m + a* are compound expressions. 29. Addition of Compound Expressions. In arithmetic if we wish to add two or more compound expressions, we write them under each other, with the Uke denominations in the same column. 38 ALGEBRA We proceed in a similar way in algebra, writing like terms in the same column. In arithmetic. In algebra. 2 yd. 1 ft. 6 in. 2a-\- h+ 6c 3 yd. 1 ft. 4 in. 3a+ 6+ 4c 5 yd. 2 ft. 10 in. 5a+26 + 10c. If the like terms are not in the same order, the}' must be properly arranged for addition. Ex.— Add 5x+3^— 22, 4;/— 52 + 3x, —32+4x4-?/. Here the problem might be written thus : 5x-l-3i/- 2z 3a; + 4!/- 5z 4x-l- y- Zz Sum=12x+8i/-102. EXERCISE 18 (1-6, Oral) Add: 1. 3 ft. 2 in. 2. 3x4- 2y 3. 5a-116 5 ft. 3 in. 5. 5x+3!/ 5a-36+2c 6. 2a- 36 4. 6 hr. 10 ruin. 11 sec. - a-36+7c 5 hr. 12 min. 3 sec. 2a— 46+3c —3a— 6+4c 2 hr. 15 min. 20 sec. 5a— h-\- c —5a— 26+ c 7* a+b—c, 26-3c+a, 36+5a— lie. 8. 5x2— 7x+6, 3_5x4-x2, _2x+4x2. 9. 2o-36, 3a— 26, 4a, —6, a— 6. 10. a + 26, 6+2c, c + 2o, a + 6+c. 11. a— 26+c— 3rf, c— 56— i+2a, — 6+c— d+o. 12. 5z—3y, —2y+z, iz—y+3x. 13. ^a-16, §a-36, io-*6, U-f^b. 14. a2+262— 3c2, 5b--c-i'2a-, 3a-+b'^—2c*. 15. a+6 + c, 6+c+d, c-]-d+a, d+a+6. 16. lx+ly-\z, Sx+Jy-fz, x+y-^z. AUDITION AND SUBTRACTION 39 30. Addition of Quantities with Unlike Signs. What is the result of combining : (1) A gain of $20 with a loss of $10 ? (2) A gain of $5 with a loss of 815 ? (3) A loss of $8 with a gain of S6 ? (4) A loss of $7 with a gain of $12 ? These might be written as problems in addition, thus : + $20 -r% b — $8 — $ 7 -$10 -$15 +$6 +S12 + $10 -$10 -$2 -8 5 It is thus seen, that when we add two quantities differing in sign, the sum is sometimes positive and sometimes negative. When is it positive and when is it negative ? How is the numerical part of the sum found when the signs are alike ? How is it found when the signs are different ? The answers to these questions might be combined into the following rule : When the signs are alike, the sum is found by arithmetical addition, and the common sign is affixed ; when the signs are different, the sum is found by arithmetical subtraction, and the sign of the greater is affixed. Ex. 1. — Find the sum of 6 and —8. Here the result is —2, since the difference between 8 and 6 is 2 and the one with the greater absolute value is negative. If there is doubt in any case, it is advisable to make the problem concrete by substituting for +6, a gain of $6 and for — 8, a loss of $8, when the result will at once be evident. Ex. 2. — Find the sum of 5a, —8a, —7a, 6a, —2a. The sum of the positive quantities is llo. The sum of the negative quantities is — 17a. The sima of 1 la and — 1 7a is — 6a, .'. the required sum is —6a. They might also be eidded in the order in which they come. Thus, the sum of 5a and —8a is —3a, of —3a and —la ia —10a of — lOo and 6a ia — 4a, of — 4a and — 2a ia - 6o. 40 ALGEBRA Ex. 3.— Add 3a- 116+ 5c, 66-5a, 56-c+a. Write the expressions in columns as already explained. a==b=c= 1 3a-116 + 5c — 5a+ 66 a+ 56— c = -3 = +1 = +5 -a +4c = +3 The sum is — a + 4c or 4c— o, the sum of the second column being zero. We may check the result by substituting particular numbers for the letters. Thus, if we substitute unity for each letter the first quantity becomes 3—11+5 or —3, the second is +1, the third is +5, and the sura (— a+4c) is +3. Since the sum of —3, +1 and +5 is +3, we assume that the work is correct. Add: 1. +6 ft. -3 ft. EXE3RCISB 19 (1-12. Oral) -3 7 9. 3a-26 5a + 36 10. -$10 + $27 3. 7. 11. + 10 lb. -151b. 4. 8. 12. -50° +40° 5a* -3a2 - 7z 13x -8a6 Sot 2x-ly -3x+2y a+66 a- 66 3a+26— c 2o— 26+c 13. 3x+5y, 2x—Zy, —4:X—y, 6ar— 4t/. (Check.) 14. 5m, —6m, —7m, Sm, —9m, 10m. 15. 2a+36— 5c, 6a— 46+c, 3a+26+4c. (Check.) 16. 3a— 56, 46-3c, 4c— 3a, a+6+c. (Check.) 17. \x+y-\z, ^x-\y+lz, x+\y+z. (Check.) 18.* a— 26 + 3c, 6— 2c + 3d, c — 3d+2a, 6— 2c— 3a. 19. 3x+5y— 2z, 2z— 3?/+42, 4z+i/— 5z, 6x+2i/+3z. 20. 3a6— 4ac+56c, 5ac—^bc — 2ab, Zbc—ab—ac. 21. 6o2-5a6+62, 2a'^-\-lab-2b^, a^-ab+b\ 22. 2x2-3j/2+4z2, 5i/*-62«+a:*, 2z*+2y«-3x«. 23. ia-Ji+Jc, 3f -JK>-4«. ADDITION AND SUBTRACTION 4t 24. If the sum of 13x— 7, 2x+5 and 6— 4a; is 48, find the valne of X and verify. 25. If the sum of i— 6, 3a;— 6 and 5a;— 6 is the same as the sum of 12— z, 12— 3a; and 12— 6a;, find x and verify. 31. Indicated Additions. If we wish to use the sign of addition to indicate that b is to be added to a, we write it thus : a-4-6. Similarly, if we wish to indicate the sum when — 7 is added to 11, we write it ll-!-(— 7), the negative quantity being enclosed in brackets. To find the value of ll+(— 7), we must add 11 and —7, which is done by subtracting their absolute values and prefixing the positive sign. Thus, ll+(-7)=ll -7 = 4. Similarly, 6a + ( — 3a) = 6a — 3o=3a, 5w+( — 3m) + ( - m^ = 67w — ?m — w=m, and o+{ — fe)=o— 6. We thus see that, to add a negative ouaniity is the same <m to subtract a positive quantity of the same absolute value. If we wish to simplify a quantity hke (3o-26)+(2a-36), we may write 2a— 36 under 3a— 26, and add in the usual way, or we may remove the brackets and say that tht quantity ==3a— 26+2a— 36, =5a— 56, when the like terms are collected. EXERCISE 20 (1-12. Oral) Simplify : 1. -3+4. 2. 10+(-6). 3. 3+(-4). 4. (-2)+(-3). 5. 5a+(-4o). 6. 76+(-4fc). 7. -8a+( + 7a). 8. -5ab+(-2ab). 9. 9x^+{-Zx' 10. -p-\-(-3p). 11. (-3m)+(-8m). 12. -o+(-2a)+(-3c). 13.* 10xy-\-{-^xy)+(-Axy)-xy+(-5xy). 4i AIMEBRA 14. -6+(-26)+(-36)+(-46) + 106. 16. (2m+3») + (5m-ra)+(3m-5»). 17. (6x+3y-42) + (x+2y-2)+(y+z-7x). 18. a+{-6) + 6 + (-c)+c+(-a). 19. z+(a-26 + c) + {6-2c + a)4-(c— 2a-}r&), 20. When —20 is subtracted from 10, the difference is 30. Show that this is true by adding the difference to the quantity which was subtracted. 21. Show by addition, that when 2a—h-\-5c is subtracted from 3a— 464-3c the remainder is a— 36— 2c. 22. Solve (2x+3) + (3z-5)f(5a;-l)=57. (Verify.) 23. Solve (8x-7) + (-4a;-3) = (-5x-7)+(7a;-2). (Verify.) 32. Subtraction is the Inverse of Addition. To subtract 4 from 7 is equivalent to finding the number which added to 4 will make 7. Thus every problem in subtraction may be changed into a corresponding problem in addition. If we wish to subtract —4 from 7, we enquire what number added to —4 will make 7. We might make the problem concrete by finding what must be added to a loss of $4 to result in a gain of $7, and the answer is evidently a gain of $11. .*. when —4 is subtracted from 7 the remainder is 11. Thus, 7 less— 4=11, because —4+ 11 = 7. -10 less -3 = -7, because -3 + {-7)= -10. 66 less —46=106, because —46+106 = 66. BXBRCISB 21 (Oral) What must be added to 1. A gain of SIO to give a gain of $15 7 2. A gain of $8 to give a gain of $3 ? 3. A gain of %5 to give a loss of $4 7 ADDITION AND SUBTRACTION 43 4. A loss of $6 to give a gain of $3 ? 5. A loss of §20 to give a loss of $15 ? 6. A loss of $5 to give a loss of $8 ? 7. 8 to give 12. 8. 10 to give 3. 9. —8 to give 2. 10. — 8 to give —2. 11. 7x to give 3x. 12. 3a to give —5a. 13. 5x^ to give —2x2. 14. — 8x to give — lOx. 15. 3abc to give 2abc. 16. — 6y^ to give — 9y*. 17. —5t to give — 4<. 18. a to give —a. 33. Rule for Subtraction, tractions : Examine the following sub- (1) 9 (2) 3 9 -3 (3) - 9 (4) -9 3 -3 6 12 -12 -6 In (1) the result 6 might have been found equally well by adding —3 to 9, in (2) by adding +3 to 9, in (3) by adding —3 to —9, and in (4) by adding ~\-S to —9. Thus we see that the problems might have been re-written as problems in addition by changing the sign of the quantity to be subtracted. We would then have these problems in addition : (1) 9 (2) 9 (3) - 9 (4) -9 -3 +3 - 3 +3 D 12 -12 -6 We have therefore the following rule for subtraction . Change the sign of the quantity to be subtracted and add. To subtract compound expressions, apply the rule to the Uke terms of the expressions. Thus, to subtract 6a— 36 + 2c change to 6a — 36 + 2c and add. 3a+46 — 6c -3o-46 + 6c 3a— 76+ 8c. 44 ALGEBRA EXERCISE 22 Re-write the following problems in subtraction as problems in addition and find the result : 1. 25 -13 -16a2 -20a 2 2. -11 3. + 3 6. 56 7. 86 10. 6x-Zy 11. -2z+4y Subtract a— 26 + 3c froc Subtract 5a:2_lla;+4fr -10a - 8a 4, 8. 12. + 5. 20x - 4x 6. ix^y -3xh/ 5m+4n 5m— 4ri -23m 2 - 6m2 9. 3a +76 2a-26 6x*-5a:+2 3x«-2x-3 13 14. a 3a-46+c. om 3x^—2x As soon as possible the pupil should learn to subtract, without actually changing the signs, but b}' making the change mentally. Subtract : EXERCISE 23 (1-8. Oral) t. 8 -3 2. -9 4 3. -12 - 6 4. . -la 2a 5. —8a; -3x 6, -7m —2m 7. 11. — 4a6c llc6c 8. 12. -9i 9. 3x+4y x-2y 2a-36 3a +26 7x2-5 3x*-7 6m -3n -3m+6n 13. 2m+4n— 3j3 from 5m — Zn-r(>p, and verify the work by addition. 14. Find the remainder when 6a— 46— 5c is subtracted from 2a+36-llc. (Verify.) 15,* Subtract 5a— 36 from the sum of 2a — 6 and 6a— 46. 16. Subtract a— 26+c from 6a — 6 + 5c and from the remainder subtract 3a+6— c. ADDITION AND SUBTRACTION 46 17. Subtract the sum of 3x^— 5a;+6 and 5x*-|-4a;— 3 from 6a;^— x+3- Check when x—\. 18. What must be added to 2m+3n— 4p to give 5m--n—2p 7 (Verify.) 19. By how much is 7a*— 15a— 11 greater than 3a*— lla + 4 7 20. By three subtractions simplify (6a+106-c)-(3a+46-2c)-(a-36-[-4c)-(2a+76-3c). 21. Subtract the sum of 2p—5q—3r, —p-'r3q—2r and 4^ + 6^— 4r from the sum of 3p—4:q-'r5r, 3q—4:r-\-5p and 3r—4ip-{-5q. 22. Subtract 2a— 36+ 5c from zero. 23. What is the excess of 15 over 10 ? 8 over —2 ? —4 over — 11 ? a+b over a-2b ? 3a;2-5x+2 over 2^2— llx+7 ? 24. Add a*-|-2a— 5 to the excess of 2a^— 4a+3 over a^— 3a+10. 25. Subtract the sum of o— 36+c, 6— 3c+a, and c— 3o+6 from zero. 26. What must be added to the sum of x^—5x, 2x^— 3a:+4 and 6a;— 3x* so that the result wiU be unity ? 27. From 2z-3x^+o-x^ take 3 — Ux^-5x^—6x. 34. Indicated Subtractions. If we wish to indicate that — 3 is to be subtracted from 9 we write it: 9— (—3). From the preceding this is at once seen to be equal to 9 + (-)-3) = 12. Also, -8-(-5)= -8 + ( + 5)= -3. — 3o — ( — 5a)= — 3a + of = 2a. o-(-6)= a + ( + 6) = a + 6. We thus see that to subtract a negative quantity is the same as to add a positive quantify of the same absolute value. Similarly, (3o + 4fe)-{2a-36) = (3a + 46) + (-2o + 36), = 3a + 46 — 2a + 36, =a+76. Also a—(b+c)—a-\-{ — b — c) — a—b—c, and a— (6 — c) = a + ( — 6 + c)=a— 64-c. Thus, brackets which are preceded by a minus sign may be removed if the signs of all the quantities within the brackets be changed. an ALGEBRA Ex.— Simplify 5x— 3?/+42-(3.r— 2?/4-2z). 5x-3y + 4z We may consider this as an ordinary problem 3x—2y + 2z in subtraction and proceed in the usual way. 2x- y + 2z We may, however, remove the brackets using the rule and then collect the like terms, thus : The expression =5x — Zy + 4z — 3x + 2y — 2z, = 2x—y + 2z. EXERCISE 24 (1-10. Oral) Simplify : 1. 10-(-3). 2. -5-(-6). 3. -7a-(-4a). 4. 8z-(-3x). 5. (-2m)-{-3m). 6. -{-b)+b. 7. 8-(-4)-(-2) 8. 8ab-l0ab-{-lab). 9. ??i-(-3m)-(-5m). 10. -4:X^+(-3x^)-{-lx^). 11.* {5x-2y)-{2x—4.y). 12. .3a-116-(5a-86). 13. 2a-36+5c-(a-46+5c). 14. (a+6)-f(2a-36)-(4a-36). 15. a-rb—c—{b-\-c—a) + {a^b—c). 16. (6x^—3x+5)+(2x^—5x—6)—{5x^—8x+2). 17. Find the value of 5a-\-b, when a = 2, 6 = — 3. 18. Find the value of 2a+3b—c, when a = l, b^2, c=— 3. 19. By two different methods find the value of a- (b—c), whei 0=20, 6=10, c=7. 20. Solve and verify : (1) 2a;-3-(x-4)=-8. (2) 3x-l-(a:-3)-(z+7)=40. (3) l-(4-x)-(5-x)-(6-x)=52. 21. What value of x will make 5x— 6 exceed 3x— 11 by 70 ? 22. Find the value of 3x*— 2*+5— (2x«+x— 1), when*=0, 1, 2, 3, 4. ADDITION AND SUBTRACTION 47 23. Remove the brackets and simplify : (1) (a+36-llc)-(6 + 3c-8a)-(c+5a-26). (2) (a-36)-(6-3c)+(c-3d)-(d-3a). (S^ —{Zx—y+2z)—{2x—Zy+^) — {'iy-%z—bx). (4) -{a-h)-{b-c)-{c-d)-{d-a). E3XERCISB 25 (Review of Chapter TV) 1,* Find the sum of 5a, —3a, la and —8a. 2. Find the sum of four consecutive integers of which x is the least. 3. Find the sum of five consecutive integers of which n is the middle one. 4. Add 3a-26 + 7c, 56 — 3c — 2o and c-a-36. 5. Subtract —106 from —66, —3a from 5a. 6. From 4a — 36 -f 5c subtract 2a — 56 — c. 7. Subtract 5x— 3t/ + 42 from 4:X—2y — z. 8. What must be added to 3a — 56 + 6c to give 6a — 76 + 4c ? 9. If x-\-y—\Q and x — j/ = 4, what is the value of 2x ? What is the value of 2t/ ? 10. What is the sum of the coefficients in 6a— 116 + c — 3d ? 11. What must be added to x — j/ to give ? 12. When x=3 and i/ = 4, what is the remainder when x* — y* is subtracted from 2xy 7 13. What is the difference between 2a — 6 — c and a — 6 + c 7 Give two answers. 14. To the sum of 3m — 4n and 2m— 3n add the sum of m+7n and 3m -2n. 15. From 4x* + 3x— 7 subtract the sum of 2x* + 7x— 5 and 2x»-8x+7. 16. By how much is 3x— 7 greater than 2x+5 ? For what value of X would they be equal 7 17. Simplify a + (26 — 3c)-(c + a). 18. If x = a+26 — 3c, i/ = 6 + 2c — 3a and z = c + 26-3a, find the value 0f(l)x+(y + 2), (2)x-(j/-2). (3)x-(2/ + 2). 48 ALGEBRA 19. From the sum of •bx-\--^y and Zx—^y subtract the sum of ■2x—\-\y and Ax-\-\-Zy. 20. By how much is 3x* — 5a;4-ll greater than 3a;* — 8x+17 7 What is the meaning of the result when x = 2 7 when x = 1 7 21. From %a—^b-\-\c take Ja— J6 — Jc. 22. If a + 6 + c = 0when a=3x — 4;/ and 6 = 4)/ — oz, what is the value of c 7 23. Solve 5x-3-(x-4)-(x-2) = 27. (Verify.) 24. What value of x will make 3x— 2 exceed x— 7 by 63 7 25. Whena=l, 6 = 2, c = 3, the sum of x-f-a — 36 + 4c, 2x + 6-3c+4o and 3x + c — a — 6 is 124. Find the value of x. 26. Using breickets, indicate that the sum of a and b is to be diminished by the sum of c and d. If a = 2x — 3, 6 = 5 — 3x, c = 3x— j, d = I — 6x, what is the result ? 27. Solve 2-(x-i)-(i-3x) = 16-25. 28. Subtract 2m— 7n — 4x from zero. 29. What must be added to a — (1— 6) — (1 — c) to produce unity 7 30. Subtract the sum of 2a -|- 36 — 4c -j- d and a — 26 -c — d from the excess of 4a — 6-j-c over a + 6-t-c. CHAPTER V MULTIPLICATION AND DIVISION 35. Multiplication of Simple Positive Quantities, of a product may be taken in any order. Thus, Also 3x5x4 = 3x4x5 = 5x4x3 = etc. axbxc=axc X 6 = 6 xaxc =etc. The latter product may be written abc, acb, etc 3ax26=3xax2x6, = 3x2xax6, = 6ab. Make a diagram to show that ixx2y = 8xy. Similarly, 3a6 x 5cd — 3xaxbx5xcxd, = 3x5xax6xcxa, = I5abcd. The factafs ai ab ab ah ab ab Thus, the coefficient of the product is obtained by multiplying the coefficients of the factors, and the literal part of the product by multiplying the literal parts of the factors. 36. The Index Law for .i/lultiplicatiouo a^xa^ —axaxaxaxa =a^. 2zx3x^=6xxxxxx =i5x''*. m* X m^^m .m .m .mxm .m . m=m^ Similarly, 3t/* X4y^=l2y^, and x'xx*xx^ = x^+*+^ = x^*. Thus, the index of the product of powers of the same quantity is found by adding the indices of the several factors. flO ALOBBRA BXBRCISE 26 (Oral) Find the product of : 1. 2x, 3y. 2. 4m, 5n. 3. fz, 4y. 4. 3x, ix. 6. 6a;, ^x. 6. 3a 6, 4x!/. 7. a-, a. 8. 32/^ 2y\ 9. ab, ac. 10. 2x2, 43-3, 11. 5p-, 4p«. 12. 5x2, 33^ 13. (ix^r-- 14. 3ax, 2ax. 15. <2, t\ t*. 16. ab, ac, a. 17. 5a2, 3a, 2. 18. (4a)3. 19. la, ib, Ic. 20. (36)2, (26)2. 21. §w, 6ra, dmn. 37. Multiplication by a Negative Quantity. 4x3 is a short way of writing 4+4-J-4=12. —4x3 is a short way if writing (— 4) + (— 4) + (— 4) = — 12. Hence multiplication by a positive integer means that the multiplicand is taken as an addend as many times as there are units in the multipher. Also, we shall define multiplication by a negative integer as meaning that the multiplicand is taken as a subtrahend as many times as there are units in the multiplier. According to this definition then 4x-3 = -4-4-4=-12, _4x-3=-(-4)-(-4)-(-4), = +4+4+4= + 12. We may state these results in algebraic symbols, thus : (+a)y(4-&)= + a6, ( — a)x( + 6) = — a6, {-\-a)x( — b)=—aO, {—a)x( — b)^-\-ab. 38. Rule of Signs lor Multiplication. Examine the preceding statements and state when the product has a positive sign, and when it has a negative sign. The rule of signs for multiplication may be stated in the form : The product of two factors with like signs is vosilive, and of two factors with unlike signs is negative MULTIPLICATION AND DIVISION 61 EXERCISE 27 (Oral) State the product of : 1. 6, -7. 2. -4, 3. 3. -4, -5. 4. 3, —5. 5. 3x, —2. 6. —X, y. 7. —X, —2?/. 8. 2t7i, —3n. 9. — x, —3x7/ 10. -o(-6). 11. (-6)2. 12. -ayi-x). 13. — 2?ni;( — v). 14. —\x,—\2x. 15. — 2.r2x— Sx" 16. x2, —X. 17. — 2a2, —3a. 18. — ofcx— 3crf. 19. — oxy, 2x2. 20. — a^ — a. 21. 3a%X—ah. 22. 5x32/3, — X2/2. 23. Za^b^c^, —abH. 24. —x'^yz^, 5x^yh. 25. Since —4x3 = — 12, —4x2=— 8, —4x1 = — 4, what would fou expect —4x0 to be equal to ? Also — 4x— 1 and — 4x —2 ? 39. Multiplication of several Simple Factors. Ex. 1.— Multiply 2a, -36, —4ab, —b. The product of 2a and — 36 is — 6ab. „ „ „ —6ab and —4ab is 24a*6*. „ „ „ 24a«6« and -& is -24a26S .*. the required product is — 24o'6^. Of course the factors may be multiplied in any order we choose. If we multiply all the negative factors first, what sign will their product have ? What sign will the product have if we multiply four negative factors ? Five negative factors ? Twenty negative factors ? The product will be negative vvhen the number of negative factors is odd and will be positive when the number of negative factors is even. Any number of positive factors will evidently not affect the sign of the product. Ex. 2.— Multiply 3x, —5xy, -6x2, _y^ QyS (1) The sign of the product is — , since there is an odd number of negative factors. (2) The numerical coetficient = 3 x 5 x 6 x 6 = 540. (3) The literal part of the product = x. xy . x* .y . y^ — x*y^, ,", the complete product = — 540xH/*' E 2 52 ALGEBRA Ex. 3.— Find the values of (-2)3, {-2)^ (-2)^ In ( — 2)* the number of factors is odd, /. (_2)3=-2»=-8. Similarly, (-2)*= 16 and (-2)6= -32. EXERCISE 28 (1-18 Oral) Find the product of : 1. 3, 4, -5. 2. 3, -4, -5. 3. _3, _4, _5. 4. —a, —b, — c. 5. 2a, 3a, —4a. 6. -i, -h 12. 7. 3x, —2xy, —y. 8. -3, -3, -3. 9. (-4)-^ 10. -2, -3, -4, -5. 11. (-2)2x(-3)2. 12. — 2x, — 2x,— 235 13. —a, —2a, —3a, - -4a, —5a. 14. 2x, —3x, 4y, — y- 15. 5xy, —Sxy, — 2x, -2y. 16. —1, —2, -3, -i. -i 17. What is the square of —2a, of —3xy, of — 4a^6c ? 18. What is the cube of —5, of —x, of —2x- ? 19.* If x= — l and ?/=— 2, find the values of : x', y^, x^+y^, x^—y^, x', y^, x^-\-y^, x^—y^. 20. Find the value of 3x^4- 2x— 5 when x=— 2; when x=— 3; when x=— 4. 21. Write without the brackets : {-a)\ {~a)\ (-2)', (-1)7, (-1)8, (-3)*, (-l)'x(-2)5. 22. Find the sum of the squares of —2, —3, —4. Find also the square of their sum. 23. Find the value of (a-6)2-f(&— c)='-l-(c— a)^ when a=:6, 6=4, c=2. 24. When o=2, 6=1, c=— 3, show that a3-|-63-f-c'=3a6c. 25. When x=3 and y= — 2, how much greater is (x— y)^ than »'— y' ? If a= — 1, 6= — 2, c=— 3, d=— 4, find the value of: 26. 3a + 26+c-4d. 27. a^+fc^+c^+d*. 28. a6+ac-l-6c-fcd. 29. a^d^—b^c\ 30. aJb€-\-bsd+cda+dab 31. o»+H+c»+rf». MULTIPLICATION AND DIVISION 53 40. Compound Multiplication. We multiply a compound quantity by a simple one in a manner similar to the method in arithmetic. In arithmetic. 3yd. 1ft. 4in. 23 = 2.10+3 2 2 2 6yd. 2ft. Sin. 46 = 4. 10 + 6 In algebra. 3a+46- 5c 2 60+86— 10c If we wish to indicate the product of x and y-\-z, we write it in the form x{y-\-z), which we see is equal to xy-\-xz. The diagram shows how this product may be illus- trated geometrically. ^ Make a similar diagram to show that a(6 + c + d)=o6+oc+od. Similarly, x{y — z)==xy~xz. Can you see that the diagram is a geometrical '^ illustration of this 7 Ex.^implify 3(a-6)-4(6— c)-2(a-6+c). The expression = (3a- 36) -(46 -4c) -{2a -26 + 2c), = 3o-36 — 46 + 4c— 2a+26 — 2c, =o-66 + 2c. y z xy KZ y-a B x(y.z) XZ EXERCISE 28 Copy and supply the products : 1. 2o+6 2. 3o— 26 4 7 3. 2in—5n -6 4. 4x— 3y 2x 5. 3x-4y -22/ 8. 3{5x-2y). 6. 9. 2a+56-c —a 7. 2(3x-ll). -2{3x-y). 10. 3x(a;2+5x-2). 11. 5xy{2x^-xy). Simplify : 13.* 3(o+6)+4(6+c)+5(c+a). 12. Zmp{5—mp). 14. 2(x-2y)+3(x- -y)-(ix-3y). 16. 3(2TO-3n)-5(m-n)+2(m+2n). 54 ALOEBRA 16. 4(o-26+c)-3(6-2c+a)-2(5c-4a-56) 17. i(2a-36) + i(2a+56) + i(5a-f&). 18. x(a:-l)4-2x(x— 3)+3x(x4-5). 19. a(a2-a+l)+3(o2+o-2)-2(a2+2a-3). 20. 3x(a;2-2x+2)-2a;(3x2+4a;-5)+x(4a;2+5x-6). 21. — 2a(6— c+d)-3a(c-d+6)— a(d-6-c). Solve and verify : 22. 3(x-l) = 2(x+4). 23. 5(x-2)-2(x+2)=70. 24. 6(2x-3)-3(x-3)=0. 25. 2(5x-9)4-4(x-ll) = 36. 26. 3(x+2)+5(x-3)=2(x-4)+4(x-l) + 13. 27. Find the sum of x(x+l), 3x(x— 2), 2x(x— 5). 28. Subtract a(2a2— a+1) from 2a(a2+3a-2). 29. If a stands for x^^xy-^-y"^ and 6 for x^—xy-\-y'^, find the values of a—b, 2a-\-b, 3a— 26. 41. Multiplication by a Compound Quantity. The measures of the sides of the large rectangle are a-\-h X y and x-\-y. The measure of the area is the product of a-\-h and x-\-y, which is seen to be ax-\-ay-\-hx^hy, :. [a-[-h){x-\-y)=ax-{-ay-\'hx-^hy. This diagram shows how to find the product of x + 3 and x + 2. What does it show the product to be ? Make a similar figvire which will show the product of a-f-6 and a-\-b, and thus find the value of (a + 6)*, or the square of a + 6. The method of obtaining the product without the diagram is similar to that used in arithmetic. ax ay bx by x^ 3x 2x 6f In arithmetic. 12 = 23 = 1. 10+2 2. 10 + 3 12x 3= 36= 3. 10+6 12x20 = 240 = 2. 10« + 4. 10 12x23 = 276 = 2. 10» + 7. 10+6 In algebra. x+2 2x+3 3x+6 = 2x* + 4x = 3(x+2) 2x(x+2) 2x« + 7x+6 = (2x+3)(x+2) MULTIPLICATION AND DIVISION 50 Thus, (he product of any two expressions is obtained by multiplying each term of the multiplicand by each term of the multiplier. The proper signs are attached to these partial products, and the sum of the partial products is then taken. In multiplying in arithmetic we begin at the right, but in algebra it is usual, but not necessary, to begin at the left. Ex.— Multiply (1) 2a-3b by 3a-26. (2) 3x-5y by 4x+y. Check = 6=1 1 1 1 (1) 2o - 36 3o — 26 Check 6a*- 9ab 4a6 + 66« 6o*-13a6 + 66» x=y = \ -2 5 -10 (2) 3x — 5y 4a; + 1/ I2x^-20xy + 3xy-5y- 12x*— l'Jxy — 5y* 42. Checking Results. In Chapter II. we saw how U verify the root which we obtained in solving an equation. We might verify our work in subtraction by addition. As in addition, the work in multiplication is easily checked by substituting particular numbers for the letters involved. Thus to check the work in the first example in the preceding article, we might substitute 1 for each letter involved. Whena = 6=l, 2a-36 = 2-3= -1, 3a-26 = 3-2 = l, and 6a*-13a6 + 66* = 6-13 + 6= -1. Since the product of — 1 and 1 is — 1 , the work is likely correct. A convenient way of exhibiting the test is shown. Of course any numbers might be used in checking, but we naturally choose the simplest ones. EXERCISE 30 Find the product of the following and check : 1. x+3 2. 2x+7 3. x+5 2+4 x+1 2x+2 4. 3a;+4 2x+3 M ^/v<7^5flyi 6. a-3 6. a— 5 7. 6-4 8. 3a— 5 a-4 2z-3 10. a+3 26-3 12. 2a+5 9. 11. 2x-3a 3a— 7c 2x+Z X-J/ 5x— a 3a4-7c 13. (3a+46)(2a-56). 14. {x-5y){2x-^ly). 15. (a+6)(2c-d) 16. (a-36)(3c-4d). 17. Find the square of x—y by multiplying it by x—y. What is {x—y)^ equal to ? 18. Find the squares of 2a— b, 2a— 36, 4a+5, 3a+46. Check by putting a=3, 6=1. 19.* Simplify (x+l)(a;+2)+(x-2)(x+3). 20. Simplify 3(a+2)(a-2)+2(a-5)(a+l). 21. When Sx^- 2x— 15 is divided by 2x— 3 the quotient is 4x+5. Prove that this is correct. 22. Show that (6x— 8)(2x-S)=(4x-6)(3x-4). 23. m{x-\-y)=mx-{-my. Find the value of mx-\-my when to=2-14, x=43-7, 2/=56-3. 24. If a train goes 2a— 36 miles per hour, how many miles will it go in 2a +36 hours ? 25. Simplify {x+y)^+{x-y)^ ; {x^y)^-(x-y)\ 26. Simplify 2(a-6)(2a + 6)-3(a + 6)(a- 26). 27. Subtract (x+2)(x— 9) from .;x+3)(x+4). 28. Multiply 3(x+3)-2(x+4) by 2(x-5)-(x-3). 29. Subtract (x4-3)(x+7) from (x+l)(x+ll). For what value of x are these quantities equal ? (Verify.) 30. Show that there is no value of x which will make (x— 10)(x— 1) equal to (x— 3)(x— 8). 31. Subtract the sum of (3x4-2)(2x+3) and (3x-2)(2x-3) from the sum of (4x+3){3x+4) and (4x-3)(3x-4). 32. Simplify (x-3)2+(x-2)(x+2)+(x+l)(x+5). 33. Subtract the product of 2a— 5 and 3a +2 from the product of 3a+5 and 2a— 2. MULTIPLICATION AND DIVISION 67 34. Find the sum of the squares of x+1, ar+2, z-f3. Check by putting x=2. Solve and verify : 36. {x-\-5)(x-l)={x-5)ix+2). 36. {x-l)^={x-6){x-\-2). 37. (2a;-l)(3x-l) = (x-2)(6x+4). 38. (a:+ll)(a;-2)=(a;-7)(x-l)+107. 39. a;(a;+l)+(a:+l)(x+2)=2(a;+l)(x+3). 40. (x+l)2+(x+2)2+(ar+3)2=3(x-2)2+14. 43. Division by a Simple Positive Quantity. To divide 24 by 6 is the same as to find the number by which 6 must be multiplied to produce 24. Thus division is the inverse of multiplication as subtraction is the inverse of addition. Since axb=ab, .'. ab^a=h and ai)^b=a. If we wish to divide 6xy by 2x, we must find what 2x must be multiplied by to produce 6xy. (1) 2 must be multiplied by 3 to produce 6, (2) X „ „ „ „ y „ „ xy. :. 6xy-^2x=3y. Similarly, I5abc-^3bc = 5a. A problem in division may be written in the fractional form, the dividend being the numerator of the fraction and the divisor the denominator. 24 . o6 , 6x1/ _ Thus. -«-='*. — = f>, -^ = 32/. 6 a 2x As in arithmetic, we may remove or cancel from the dividend and divisor any factor which is common to both Thus, -^r- = 4a, on removing the factors 3 and 6. gioul^y, l-^.= 3ay and ^=1^. 68 ALGEBRA 44. Index Law tor Division. Since a^xa^=a'' by the index law for multiplication, .'. a'''^a-=a^ or a^^ a^^a-, a^ \ .k .a .a .a . . a^ h .k .k .a .a Thus, the index of the quotient of powers of the same quantity is found by subtracting the index of the divisor from the index of the dividend. Thus, a;*-:-x' = x*~* = x* ; a''-^a' = a''~*=a^. Similarly, ^^^ = 5a^-*b^-^ = 5ab*. The work in division may be verified by multiplication. Thus the preceding division is seen to be correct, since 5ab^x3a-b = 15a^b^. EXERCISE 31 Copy and supply the quotients, verifN-ing the results by mental multiplication : ^ 3x1/ q 5abc „ 24mn . 25xyz 2. ab 3n 5z , 4a2 „ 42x3 - 12a* „ BSm^ra O, . b, . 7. — . o. . 2a Ix 2a2 Vimn 9. ??:!'''^ 10. i!^'. 11. i"-. 12. l^^V . ahc ' 6p^ ' ha ' \xy 13. 6xz/-i-2x. 14. lOa'-f-Sa. 15. ^miJ^^^r. 16. 10x5^-2x3. 17. 16a36^4a6. 18. ISx'j/V^Sx^t/z*. 19. 22a%^^Ua-h-. 45. Rule of Signs for Division. Since (+a)x(^i) = +aA, (-a)x(+6) =— oA, (-t-a)x(-6) ^—aJb, {—a)x{—b)=-^ab, it follows that +aZ> , , —06 , , —ab , -{-ab - — =+0, =+0, -r^=—Oy — ;r"=~°- +a — a -\-a — a When is the sign of the quotient + and when is it — ! What then is the rule of signs for division ? Compare it with the rule of signs for multipUcation (art. 38). MULTIPLICATION AND DIVISION S& Ex. — Divide —XOx'^y^ by — 2xy'^. (1) What is the sign of the quotient ? (2) What is the numerical coefficient ? (3) What is the hteral part ? (4) What is the complete quotient 7 EXERCISB 32 (Oral) Perform the indicated divisions : 1. 12+-3. 2. -12+-4. 3. -10+2. 4. —7+7. 5. —2a-. — a. 6. —12 + 2^ — 3. 7. 0^5. S. 0^ — 5. 9. 6a2^ — 2a. 10. ah-. — a. 11. axy-. — x. 12. 45^ — 5-i — 3. 13. lOa*-^ — 2a2. i4. -6a3+3a. 15. 27a;*^ — ^x\ 16. —\2m^n-. — 6mw. 17. x^y^z-. — xyz. 18. —40^ + — 2a'. 19. ~y V 20. ?^*!^ 21 ^'^^^ —pq —4imn^ —2x 22. FUl in the blanks in the following : (1) (2) (3) (4) (5) Dividend: 6a^ — lOa;^ — lOaftc 35m^n Divisor : 2a- — 5c — 5m Quotient : — 2x 2ac 6a 46. Division of a Compound Quantity by a Simple One. If we divide 6 ft. 4 in. by 2 we get 3 ft. 2 in., or 12 lb. 6 oz. by 6 we get 2 lb. 1 oz. Similarly, 3)9 ft. 6 in. 3 ft. 2 in. 4)16 lb. 8oz. 4 lb. 2 oz. 2)6 tens +8 units 3 tens + 4 units 3)9/+ 6i 3/+2i" 4)16a+86 4a+26 ' 2)6<+8 3«+4 a)ab+ac 6+ c 3a;)6x3-3a;» 2x^- X -a6)3a*6*-2a6 -3a6 +2 Thus it is seen, that icc divide a compound expression by a simple one by dividing eacJi term of the dividend by the divisor attaching the proper sign to each term of the quotient. to ALGEBRA BXBRCISE 88 (1-16, Oral) Divide the first quantity by the second : 1. 9a 2+ 6a, 3. 2. Qx^-\-<ix^~2x, x. 3. ISz^-lOx, Sr. 4. 16m'— 4m, 4m. 5. x^y+xy^, xy. 6. 12a2— 4a6, — 2o. 7. —ax-\-ay, —a. 8. a*-\-a^—a,a. 9. -6x2— 4xy, 2x. 10. — 6aA— 6o, —3a. 11. 60'- Sa^-f 4a, —2a. 12. a^h^—a^h'^, ab^. 13. —5a*— 10a', —5a\ 14. —4a: -(-10x2 —ex', _2a;. 15. 3y^—2y^, \y. Simplify : . 3i-|-6 lOx— 15 ._ oA-f-oc , hc-\-ah , ac+bc ID. — \- • 17. 1 -| • 3 5 a b c 18* g'+3a Sa'+6a ^^ (x+2)(a;-2)+(a;-2)(z-4) o 3a ' 2 * 20. a:«-f-xy ^ y'-xy ^^^ (a+2)(a-h3)-(a-3)(o-2) X y * 2a ' 22 ab—ac , be— ah a^—a^_^a^—a. -b a 24. Subtract (x+3)(x— 8) from (2x-4)(x4-6) and divide the remainder by x. 26. Solve and verify ^^-10^ ^ 3x2+15x _^ lOx-lS^^^^ X 3a; 5 EXERCISE 84 (Review of Chapter V) 1. State the rule of signs for multiplication and for division. 2.* If a = 3 and 6= —4, find the values of : a«, 6», ab, a* + b*, a*-b*, a», 6», a«-6». 3. What are the values of (-1)», (-lj% (-1)^", (-2)*, (-3)»t 4. Simplify 3a»x -46»x —2o6^ Safes. 5. Simplify 2a(a + 3)-f-3a(2a-5). 6. What is the area in square feet of a rectangle which is (a-\-b) feet long and (a — 6) yards wide 7 7. Make a diagram to show that 3xx4a;=12z'. MULTIPLICATION AND DIVISION 61 8. A merchant bought a pieces of silk at 60 cents a yard and b pieces at 80 cents a yard. If each piece contained 50 yards, find the total cost in dollars. 9. To the product of 3x — 2 and 2x — 3 add the product of 3a; +2 and 2a; + 3. 10. From the product of ox — Zy and 2x+y subtract the product of 3x — 2y and 2x — 3t/. 11. Make a diagram to show that the product of a + 3 and a+1 is a* + 4o+3. 12. Divide 4o' — 6o* — 8a by —2a and verify. 13. To the square of 2?n— 3/. idd the square of 3m — 2n. 14. Prove that when I5x* — 8xy—l2y* is divided by 5x—6y the quotient is 3x + 2y. 15. Find the product of a — b, a + b and a*-\-b*. Check by sub- stituting 3 for a and 2 for b. ^^ ^. ,., 4x3-8x»+12x , 15x»+10a;»-15x 16. Simplify J ^ ' 4x 5x 17. Solve (2x + 3)(3x+2) = (6x-l)(x + 3). (Verify.) 18. Simplify (2a-36)(a + 6) + (a-6)(3a + 6). 19. What value of x will make (x+3)(x + 9) equal to (x + 5)(x+6) T Could (x+3)(x+9) be equal to (x+4)(x+8) ? 20. Find the sum of (a-l)», (o-2)* and (a-3)». 21. Subtract the product of 2x~3y and 3x + 2y from the product 3x— 47/ and 4x+3i/. 4a'+2a 22. Simplify ^^ +(3-|-2a)(l-a). 23. Find the value of 2x*+3x— 1, when x= —3; when x= —4. 24. Find the product of x— 2, x+2 and x* + 4. 25. If x=o* — 3a+l and y = 2a* — a—l, find the values of 2x + 3y, *x-2y,'-±y. 26. If x=2a-f-6 and y = a-f 26, find in terms of a and b the values of ax— by 4x— 21/ x*—y* ~2 ' 3o~' 3"~' 27. If x=36-2c and y = 2b-3c, find the value of (2x—y){3x-2y). 28. If x=2, 2/= 2, 2= -4, find the value of x*+y*+z*-3xyz. CHAPTER VI SIMPLE EQUATIONS {continued from Chapter 11.) 47. Definition. An equation is the statement of the equality of two algebraic expressions. Thus, 2.r+3=13 is an equation, and the solving of it consists in finding a value of x which will make the statement true. The beginner should clearly see the difference between the value of X in an expression like 2x+3 and the value of x in an equation like 2a:;+3=13. In the expression 2a:+3, x may represent any number, and for different values of x the expression has different values. But in the equation 2.r+3 = 13, x can not represent any number we please, but some .particular number, in this case 5, which when substituted for x will make 2x-\-3 have the value 13. 48. Identity. The statement 4(x— 2)=4a;— 8 is an equation according to the definition we have given. If the first side of this equation be simplified by multi- plication, we obtain 4a;— 8, which is identically the same as the second side. It is at once seen that this equation is true for all values of x. An equation which is true for all values of the letters involved is called an identical equation or briefly an identity, while an equation which is true only for certain values of the letters involved is called a conditional equation. The usual method, however, is to call all conditional equations .simply " equations," and all identical equc^^ons, " identities." SIMPLE EQUATIONS 93 Thus, 5a; — 2 = 3x+10, is an equation, and (x+3)(a;— 3)=x* — 9, is an identity. We cannot always see mentally whether a given statement is an equation or an identity. Thus, (x+2)(x + 3) = (a;-l)(x-3) + 3(3x+l) might appear to be an equation, but if we simplify each side, we find that each becomes x*-\-5z-\-6; and this statement is therefore an identity. BXBRCISEI 85 Which of the following statements are equations and which are Identities ? 1. 8(x+3)=4x+4(a;+6),. 2. 3x{x+l)=x{x+l) + 2x{x+5)+lQ. 3. (a:-3)2-5=x(x-6)+4. 4. (2x-4)(a;-5)+(x-2)(x-3) = (3x-2)(a:-7)+40. 5. (x-\-a){x^-\-a^)=x^-\-ax{x-^a)-\'a^, 6. (a;+2)(a;-3)=a:(z+5)+3(a;-l). 49. Transposition of Terms. In Chapter II. the method of solving easy equations was dealt with. The method depended almost entirely on the proper use of the four axioms of art. 15. The following examples will show how the methods of Chapter II. may be abbreviated. Ex. 1.— Solve 7x-6=4x-f 12. Add 6 to each side, 7x = 4a; + 1 2 + 6. Subtract 4x from each, 7x — 4x=12 + 6. Collect terms on each side, 3x= 18. Divide each side oy 3, x=6. Here we added 6 to each side with the object of causing the — 6 to disappear from the first side of the equation, so that we might hs-ve only unknown quantities on that side. But the addition of 6 to the second side caused +6 to appear on that side. We might say then, that the —6 was transposed from the first side and written on the other side with its sign 64 ALOBBRA changed, and similarlj', that the 4a- was transposed from the second side to the first, with its sign changed. We therefore have the following rule : Any quantity may be transposed from one side of an equation to the other if the sign of the quantity he changed. Using the rule, the solution of Ex. 1 might appear thus : 7x-6 = 4x+12. Transposing terms, 7a; — 4x = 1 2 + 6, .-. 3x=18, .-. x=6. Verify this result. Ex. 2.— Solve 2(3a:-5) + 3(a;-5) = 7(a:— 1). Removing brackets, 6z — 1 + 3a; — 1 5 = 7a; — 7. Transposing terms, 6x + 3x— 7x= 10+ 15— 7 .-. 2x=18, .•. x = 9. Verification, when x=9 : first side =2x22+3x4 = 66, second side =7x8 =56. Ex. 3.— Solve 3(i/-2)-5(i/-3) = 17. Removing brackets, 3?/ — 6 — 5?/ +15=17. Transposing terms, 3^/ — 5j/ = 6 — 15+17, .-. -23/ = 8, 8 . •• y=—2 = -^' Verification : first side = 3( — 6) - 5( - 7 ) = -18 + 35=17. Ex. 4.— Solve (2a;— l)2-(x-3)(x-2)=3(a:-2)2— 4. Here the indicated multiplications are first performed. (2x-l)* = 4x*-4x+l, (x-3)(x-2) = x*-5x+6, (x-2)i' = x*-4x + 4, 4x»-4x+l-(x*-5x+6) = 3(x*-4x+4)-4, 4x*-4x+l-x* + 5x-6 = 3x»-12x+12-4, /. 4a;»-x»-3x»-4x+5x+12x= 12-4- 1 + 6, .', 13x=13, .-. «=1. SIMPLE EQUATIONS 65 Here the product of x — 3 and x — 2 is first found and enclosed in brackets with the minus sign preceding. In the next line the brackets are removed and the signs changed. In 3(x — 2)*, the x — 2 must first be squared and the product multiplied by 3. Note. — The beginner should not attempt to perform these double operations together. EXERCISE] 86 Solve and verify : 1. 4x-4=2x+8. 2. 3x-7=8-2x. 3. 3-3a;=9-5a;. 4. 2(x-5)=:a;+20. 5. 5(y-2)=3(2/+4). 6. 10(a:-3)=8(a;-2). 7. ll(4x-5)=7(6a:-5). 8. 7x-ll+4x-7=3x-8. 9. 14+5x=9a:-ll+3. 10. 3(5a;-6)-9x=30. 11. 7(x-3)=9(a;+I)-38. 12. 5(a;-7) + 63=9a;. 13. 72(x-5)=63(5-a:). 14. 28(x+9)=27(46-a;). 15. 7(4x-5)=8(3a;-5)+9. 16. 4(x+2)=3-3(2x-5). 17. (a;+7)(a;-3)=(x-l)(x+l). 18. (x-8)(x+12)==(x+l)(x-6), 19. 20(x-4)-12(a;-5)=x-6. 20. 5(2x-l)-3(4x-6)=7. 21. (2m-o)(4m-7)=8m2+52. 22. 5(3;i+l)-7;i-3(;i-7)=6. 23. (x+5)2-(x+3)2=40. 24. (x+5)2-(4-x)2=21x. 25. 4(2i/-7)-3(42/-8)-22/-7. 26. (x+4)(x-3)-(x+2)(x+l)=42. 27. (2x-7)(x+5)=(2x-9)(x-4)+229. 28. (x+l)2+(x+2)2+(x+3)2=3(x+l)(x+4)-7. 29. 2(x-l)2-3(x-2)(x+3)=32-(x-3)(x-4). 30. What value of x will make lOx+11 equal to 5x— 9 ? 31. Prove that 3(x-2)+4(3x-5)=5(3x— 6)+4 is true for all values of x. * 32. What value of a will make 5(a— 3) exceed 3(a— 7) by 28 ? 33. For what value of x wiU the sum of 12+7x, 4a;+3 and 9—5* be wern^ F 66 AWEBRa 3-4, If a; =2 is a solution of the equation (a;+l)(x+2)=(a;-4)(x-5) + 4, find the value of k. 35. Prove that 10 is a root of the equation (x+3)(a:+4) + (x+5)(x+6)=422. 36. When (3x+2)(4x— 5) is subtracted from (-2x+7)(6a;+3) the remainder is 141. Find x. 37. What value of y will make (2/— 3)(?/+3) exceed (y+4)(y— 7) by 40 ? 38. What value of k will make (5 — 3A-)(7— 2t) equal to (ll-6it)(3-it) ? 39. What is peculiar about the equation (x-5)2-(x-3)(x-7)=0? 40. Under what condition is the square of x+3 equal to the product of X— 1 and x-|-6 ? 41. If 3(2x— 1) is greater than 12(x— 3) b}' the same amount that 5x is greater than 22, find x. 42. If 4ax— 4a2=ax+2a^ what is the value of x ? 43. The lever in the diagram is balanced by the weights P and Q when Pa—Ql. The point of support F is call d F 6 the fulcrum. If P=10 lb., ^=15 lb. and a = 12 in., what is the length of 6 ? 1 p Q 44. Two boys balance on a teeter 16 feet in length. The heavier boy weighs 85 lb. and the point of support is 6 feet from his end of the teeter. Find the weight of the other boy. 45. How far from the larger weight must the fulcrum be placed, if weights of 8 lb. and 16 lb. balance at opposite ends of a lever 12 feet long ? 46. The formula C=f,{F—32)is used to change Fahrenheit readings of a thermometer to Centigrade readings. If F=n°, find the value of 0. 47 . Change the following readings to Fahrenheit readings : 0°C., 40°C., 100°C., -10°C., -50°C. 48. What is the temperature when the two scales indicate equal numbers ? SIMPLE EQUATIONS 67 50. Equations with Fractional Coefficients. Ex. 1.— Solve lx-{-\x=2Q. Since ^ + ^ = 1, .-. gx = 20, .-. a; = 20H-| = 24. Instead of adding the fractions, we might get rid of them by multiply- ing each term of the equation by 6. Then ^xx B + ix X 6 = 20x 6, .-. 3a;+2x=120, 5x=120, a; = 24. Verify by substituting in the original equation. Ex. 2.— Solve |(a;+l)+i(a;+2)-|(a;+14). Multiply each quantity by 12 (the L.C.M. of 2, 3, 4), .-. ^(x+l)xl2 + i(x + 2)xl2 = J(x+14)xl2, 6(x+l) + 4(x + 2) = 3(x+14). Complete the solution and verify. Ex. 3.— Solve x—2 x—^_x—l ~5 6^ """To" Multiply by 30, ' .-. ~ ^ X 30 - ^ x 30 = %r^ X 30, .-. 6(a;-2)-5(a;-3) = 3(x-7), .-. 6z-12-5x+15 = 3x-21, 6x-5x-3x= 12- 15-21, .-, -2x=-24, x=12. Verification rfirstside =1^ — ^ = 2 — 1^ = ^. second side = ^^^ = J. Note. — In this solution the beginner is advised not to attempt to omit the line with the brackets. He may, however, omit the preceding line when he feels that he can safely do so. 51. Steps in the Solution of an Equation. In solving an equation the steps in the work are : (1) Clear the equation of fractions by multiplying each term by the L.C.M. of the denominators of the frad,ions. (2) Remove any brackets which appear. F 2 68 ALOSBRA (3) Tran.spose all the unknown quantities to one side and the known quantities to the other. (4) Simplify each side by collecting like terms. (5) Divide each side by the coefficient of the unknown. (6) Verify the result by substituting the root obtained in the original equation. BXBROISES 87 Solve and verify : 1. p=x+5. 2. ^x=^x-\-2. 3. ix-Jx-lO. 4. ^z + ix+ix=2Q. I 2x 5. ta;+|x=a;+5. 6. ^x = -^ - ^• 7. §2/=i2/+i 8. l + l + l^x-4. X X X ^ 3ot 7m ^ »• 2-5 = 4 + 1- 10. ^-^ = 4. 11. |-| = ^-15. 12. ix+lx=li-x. 13. | + 2=H + ^-|- 14. ix-|+7x=3a;+li. 16. |-j = 2§. 16. i(x-3)=20. 7a;+2 4x-l x+1 17. -5- = -2-- 18- ^r-^ = ^- a; a;— 8 ^ „„ a;— 1 , a;+3 !»• 3 + ^ = ^- ^^- -4- + -5- = «- 21. i(x-3)+i(x-5)=0. 22. J{x-6) = Kx+5) + i(a;-13). 3x-l 5 X 2z+l „ X4-2 , „ x+4 , x+6 23. ^r- + To = 7 + 3 ^12 "4"^ 5 • 3^5 '7 8" X— 3 2x— 4 , 3x— 5 „„ -, o\ 1/ c\ 1 26. -i- = -^ + -^i— • 26. i(2/-3)-i(y-6)=l 27. I 5 -■■ 29. 1-2 X4-2 x-3 2 ~ 4 ~ 3 31. X 5z+9 2x-9 4~ 6 ~ 6 SIMPLE EQUATIONS 69 '^**' 8 ~ 11 -"• x+1 1 2x-\ 30. -2-_- = ^_^-. x-1 x-2 3-x 33. 5(x-2) = 3-65. 34. 2-34=4(a:+l-5). 35. •ox—^ = -25x-{-'2x. 36. •2(x— l) + -o(x— 9)=3. 3x— 9 x+1 3x— 14 x+6 3x— 16 x+3 37. ~~7~-H~- 8 ■ ^*- "^~ 12 ~^~ 6 * 2— X 3— X 4— X 5— X 3 39- -^ + -4- + ^ + -6- + 4 = ^- X— 1 2— X 2x— 1 2— 3x ^' ~9 4 1^+ 30 ""• 52. Problems leading to Simple Equations. In Chapter 11. we saw how certain arithmetical problems might be solved by means of equations. The steps in the solution of such problems are stated in art. 19, to which the pupil should now refer. The beginner will find his chief difficulty with step 4, in which he is required to translate the statements given in ordinary language into algebraic language. Some examples are now given to illustrate how this trans- lation is effected. Ex. 1. — Find three consecutive numbers whose sum is 63. If we let X represent the smallest one, what would represent the others ? How woiild you now express that the sum is 63 ? We thus obtain the equation : r + (x+l) + (x + 2) = 63. Write out the full solution of this example and verify the result- 70 ALGEBRA Ex. 2. — A is 3 times as old as 5 ; 2 years ago A was 5 times as old as B was 4 years ago. Find their ages. Let X years represent B's age. What will now represent A's age ? What will represent ^'s age, 2 yea>'s ago ? What will represent jB's age, 4 years ago ? Now express that 3x — 2 is 5 times x — 4. The complete solution might appear thus : Let X years = .I's age, 3a; „ =4's age, .'. (3a;— 2) ,, =^'s age, 2 years ago, .'. (x— 4) „ =B's age, 4 years ago, . .-. 3x-2 = 5(a;-4), .-. 3x-2 = .5x-20, 18 = 2x, a; =9. .*. B's age is 9 years and A'a is 27 years. Ex. 3.— How do you represent 3% of 130 ? 4% of $27 ? 5% of §a: ? 2i% of $(x+50) ? Solve the problem : " Divide S620 into two parts so that 5% of the first part together with 6% of the other part will make $34." Let $x = the first part, $(620-x) = the other part, i§, of Sx=5% of the first part, .'• iL of S((>20-x) = 6% of the other part, .-. T-§T)a;+i^(620-x) = 34, 5x+6(620-x) = 3400. Complete the solution and verify the result. Ex. 4. — What is the excess of 73 over 50 ? What is the defect of 30 from .50 ? What is the excess of x over 50 ? The defect of x from 89 ? Solve the problem : " The excess of a number over 50 is 11 greater than its defect from 89. Find the number." SIMPLE EQUATIONS 71 Let x = the number, then X— 50 = its excess over 50, and 89 — x = its defect from 89, .-. x-50 = 89-x+ll. Complete the solution and verify. Ex. 5. — The value of 73 coins consisting of 10c. piecep and 5c. pieces is $5. How many are there of each ? Let x=the number of 10c. pieces, .-. 73-x= „ „ „ 5c. The value of the 10c. pieoes= lOx cents. The value of the 5c. pieces = 5(7 3— x) cents, .-. 10x+5(73-x) = 500. Complete the solution and verify. The pupil should be careful to express each term of the equation in the same denomination. Why would it be incorrect to say that 10a;+5(73-a:) = 5 ? SXSRCISB 88 All results should be verified. 1. A number is multiplied by 23 and 117 is then added. The result is 232. Find the number. 2. From the double of a number 7 is taken. The remainder is 95. Find the number. 3. Three times a number is subtracted from 235 and the result is 217. Find the number. 4. Five times a number with 33 added is equal to 7 times the number with 18 added. Find the number. 5. Find a number such that the sum of its third and fourth parts may be 35. 6. A has $10 more than 3 times as much as B, and they together have $250. How much has each ? 7. The sum of two numbers is 81. The greater exceeds 6 times the less by 4. Find the numbers. 8. Find a number whose seventh part exceeds its eighth part by 2. 72 /SL&EBEa 9. The excess of a number over 42 is the same as its defect from 59. Find the number. 10. Find 3 consecutive numbers whose sum is 129. 11. Divide 114 into three parts so that the first exceeds the second by 15 and the third exceeds the first by 21. 12. Divide S176 among A, B and C so that B naay have 816 less than A and §8 more than C. 13. A man sold a lot for $2280 and gained 14% of the cost. What did the lot cost ? 14. Divide 420 into 3 parts so that the second is double the first and the third is the sum of the other two. ■•6. A man buys 8 horses at $x each, 5 at ${x-\-5) each and 3 at $(a;+25) each. The total cost is $2020. Find x. 16. Find a number which exceeds 31 by the same amount that J of the number exceeds 1. 17. Find a number which when multiplied by 6 exceeds 35 by as much as 35 exceeds the number. 18. A farmer sells 7 cows and 17 pigs for $754. Each cow sells for $70 more than each pig. What is the price of each cow ? 19. If 10 be subtracted from a number, 40 more than J the remainder is 30 less than the number. Find the number. 20. Find two consecutive numbers such that the sum of J of the less and ^ of the greater is 44. 21. Divide 46 into two parts so that if the greater part is divided by 7 and the other by 3, the sum of the quotients is 10. 22. Divide 237 into two parts so that one part may be contained in the other 1 J times. 23. A horse was sold for $116-25 at a loss of 7%. What did he cost ? 24. The difference between the squares of two consecutive numbers is 17. Find the numbers. 25. A box contains two equal sums of money, one in half-dollars and the other in quarters. If the number of coins is 30, how much money is in the box ? 26. ^ is 35 years old ; fi is 7 years old. In how many years will A be twice as old as B I SIMPLE EQUATIONS IZ 27. My age in 20 j'ears will be double what it was 10 years ago. What is my age ? 28. A is 35, 5 is 7 and C is 5 years old. How long wUl it be before 4's age is the sum of the ages of B and C 1 29. Find three consecutive even numbers such that the sum of a fourth of the first, a half of the second and a fifth of the third is 17. 30. ^'s share of $705 is * of S's and 5's is f of C"s. What is the share of each ? 31. The simple interest on a sum at 2% together with the interest on a sum twice as large at 3J% is $135 per annum. What are the sums ? 32. Three % of a certain sum together with 4% of a sum which is $50 greater is $12'50. I'ind the sums. 33. The value of 52 coins made up of quarters and ten-cent pieces is $10. How many are there of each ? 34. A square floor has a margin 2 feet wide all around a square carpet. The area of the margin is 160 sq. ft. Find the dimensions of the room. 35. In any triangle the sum of the angles is 180°. The greatest angle is 35° larger than the smallest angle and 10° larger than the other angle. Find the angles. 36. The length of a room exceeds the width by 4 feet. If each dimension be increased by 2 feet the area wUl be mcreased by 52 sq. ft. Find the length. 37. If I walk m miles at 4 miles per hour and m+2 mUes at 3 miles per hour, the whole journey wUl take 15 minutes longer than if I walked at the uniform rate of 3| miles per hour. Find the length of the journey. 38. A and B together have $65, B and O have $100, C and A have $95. How much has each ? 39 State problems which wiU give rise to che following equations : (1) 5x-10=60. (2) 4a;-a;=24. (3) ?-l-?=x-10. (4) 23-5a;=4a;-4. 40. A fruit dealer buys apples at the rate of 5 for 3 cents and sells them at the rate of 3 for 5 cents. How many must he seU to gain $1-28? 74 ALGEBRA 41. The sum of two numbers is 147 and J of the less is 9 greater than \ of the other. Find the numbers. 42. John has \ as much n-oney as his brother, but when each has spent 25 cents, John has only | as much as his brother. How much has each ? 53. Algebraic Statements of Arithmetical TJieorems. If we take any two numbers, say 23 and 13, and add together their sum and their difference, we will find the result is twice the larger number. Thus, 23 + 13=36 and 23-13=10, and 36 + 10=46, which is twice 23. We see that it is true for the numbers 23 and 13, and we would find it true for other pairs of numbers, but we are not sure it is true for all pairs of numbers. By the use of algebraic s^'mbols and methods, we may show that the statement is true for every two numbers. Let the larger number be a and the smaller 6. Their sum is a + 6 and their difference is a — b. But (a + fe) + (a-6) = tv + 64-o-o = 2a, and 2a is twice the larger number. Thus the statement (a+6) + (a— &) = 2a represents in a brief form the theorem stated at the beginning of this article. Besides stating it in a concise form it shows that it is true generally. EXERCISE 39 Show that the following statements are true for all numbers : 1. The sum of two numbers is equal to their difference increased by twice the smaller number. 2. The difference between the sum of two numbers and the difference of the same two numbers is twice the smaller number. 3. Half of the sum of two numbers increased by half of their difference is equal to the larger number. 4. The sum of two numbers multiplied by one of them is equal to the square of that one, plus their product. SIMPLE EQUATIONS lb 5. The square of the sum of t.vo numbers is equal to the square of their difference increased by four times their product. 6. The sum of three consecutive numbers is equal to three times the middle one. 7. If two integers differ by 2, twice the square of the integer between them is less by 2 than the sum of the squares of the two integers. 8. Read the statement {a-^b)^-\-(a—b)^=2{a^+b^) without using symbols and prove that it is true. BXBR0ISE3 40 (Review of Chapter VI) 1. What is an equation ? An identity 7 2. What rule is followed in transposing terms ? 3. Solve and verify: 6x(2x+3) = (3x+2)(4a;+ 3). ^x 3 8x 6 4. Is — ^ — = — - — an equation or an identity ? 5. What value of x will make 5(x— 3) — 4(a;— 2) equal to zero 7 x-1 a;- 10 a;-ll 6. Solve — -- H =— = — h 2. 5 7 t> 7. The sum of two numbers is 50. If 5 times the less exceeds 3 times the greater by 10, what are the numbers ? o oi_ XI ^ , , a; + 3 5x + 6 , x—2 . ^ , ,, , 8. Show that x — 1 H -— = — 1- — — - is true for all values 6 o 2 of X. 9. What value of x will make the product of 5— 3x and 7 — 2x equal to the product of 11 — 6x and 3 — x ? 2x 3 3x 4 10. If _ = — |- -262, find x correct to two decimal places. 11. A and B invested equal sums. A gained $200 and B gained $2600. If B then had 3 times as much as A, how much did each invest ? 12. From a cask which is |ths full, 36 gallons are drawn and it is then half full. How much will the cask hold ? 13. Show that x=6 is a root of (x-l)(x-2)(x-3) = 2x(z-5)(2x-7). 14. A man has $115 in $2 bills and $5 bills. If he has 35 bills Altogether, how many of each has he 7 76 ALGEBRA 15. If — H -— = 31 and a = i, find x. 16. In a ntairway there are 45 steps of equal heights. If they had been one inch higher, there would have been only 40 steps. How high ia each step ? <-f o 1 x — 4 x—5 x—2 17. Solve ^ 6-=^- 18. Divide 150 into two parte such that if the smaller be divided by 23 and the other by 27 the sum of the quotients will be 6. 19. The difference between the squares of two consecutive numbers is 51. Find the numbers. 20. A father is 30 years older than his son ; five years ago he was four times as old. Find the son's present age. 2x4-3 a; 4- 5 21. If the sum of the fractions — ;; — and — =— is 9, what is the 3 7 numerical value of each fraction 7 22. Show that the difference between the squares of any two consecutive numbers is equal to the sum of the numbers. Show also that the sum of their squares is one more than twice their product. 23. Solve 2-(a;-4-|-3x-5)=10-x. 24. If the product of x+2 and 2x+5 is greater than the product of 2x+l and x + 3 by 127, find x. 25. Solve |(2-3a;)-|(x-4) = i-(x-5). 26. Divide -75 into two parts so that three times the greater exceeds six times the less by -75. X— 3 24-x 1 2x 27. Solve _+-f -._=0. 28 . A man walked a certain distance at 3 miles per hour and returned by train at 33 miles per hour. His whole time was 4 hours. How far did he walk ? 29. Prove the accuracy of the following statement : " Take any number, double it, add 12, halve the result, subtract the original number, and 6 will remain." 30. Solve| + ^^-^^=x-8. 31. How many minutes is it to 10 o'clock if three-quarters of an hour ago it was twice aa many minutes past 8 ' 32. What value of a will make 2(6x+o)-3(2x+a) = 4(lix-6) «n identity T SIMPLE EQUATIONS 77 33. Solve (6x-2)(2x-l)-(4x— 2)(3a;-2) = 4. 34. A rectangular grass-plot has its length 5 yards longer than its width. A second plot, of equal area, is 5 yards longer and 3 yards narrower than the first. Find the dimensions of the first. 35. Solve (a;+l)(x+2) + (x + 3)(x4-4) = 2x(x+12). 36. A man leaves his property amounting to S7500 to be divided among his wife, two sons and three daughters. A son is to have twice as much as a daughter, and the wife $500 more than all the children together. Find the share of each. „ „ , x-2 , 4x+5 7x-8 ^ 37. Solve— +-g g- = 0. 38. Find an integer whose square is less than the square of the next higher integer by 37. _- „ 2x+l , 3x-2 , x-2 - , 39. If — - — exceeds — - — by - „ , find x. 3 4 -^ 6 40. How far can I walk at 3 miles per hour and return on a bicycle at 10 miles per hour and be absent 6 hours 4 minutes ? 41. A man invested f of his money at 3%, J at 4%, ^ at 5% and the remainder at 6%. If he receives an annual income of $516, how much did he invest ? 42. Prove that the product obtained by multiplying the sum of any two numbers by their difference is equal to the difference of their squares. CHAPTER \T1 SIMULTANEOUS EQUATIONS 54. Equations with two Unknowns. The sum of two numbers is 10. What are the numbers ? It is evident that there are many different answers to this problem. The numbers might be 1 and 9, 2 and 8, 3 and 7, etc., or J and 9|, — 3 and 13, etc. If we are also given that the difference of the numbers is 4, then only one of these answers will satisfy this new condition. The numbers would evidently be 7 and 3. If we follow the method previously adopted and represent the required numbers by x and y, where x is the greater, the first condition would be expressed by the equation x+?/=10. As stated, any number of pairs of values of x and y will satisfy this equation. If the second condition be expressed in terms of the same unknowns, we have another equation x—y=4:. It is now required to find a pair of values of x and y which will satisfy x-\-y=lO, and x—y= 4. If we add the corresponding sides of the equations we get : 2x=14, /, x=l and /. y=3, .'. 7 and 3 are Mxq required numbers. SIMULTANEOUS EQUATIONS 79 55. Simultaneous Equations. Any equations which are satis- fied by the same values of the unknowns are called simultaneous equations. Thus, x—1, y=3 satisfy both of the equations x-\-y=\0 and x—y=4:. To find a definite pair of values of x and y it is seen that we must have two equations containing these letters. To solve any problem where two numbers are to be found we must have two conditions given, from which the required equations may be obtained, Ex. 1.— If 5 men and 4 boys earn $43 in a day, and 3 men and 4 boys earn §29 in a day, what sum does each earn in a day ? Why do the first set of workers earn more than the second ? How much more do they earn ? -How much then does one man earn ? How can we now find how much a boy earns ? We might solve this problem algebraically, thus : Let $a; = the wages of a man for a day, and $2/ = the wages of a boy for a day. The conditions of the problem would now be expressed algebraicall}- by the equations : ' $5a;+$42/=$43, $3a;-|-$4y=$29. Or, omitting the $ sign and using only the numbers, 5x + 4?/ = 43, Subtract the terms of the second equation from the corresponding terms of the first, .-. 2x=14, .-. x= 7. Substitute x=7 in the first equation and 35 +42/ = 43, .-. 41/ = 8, .-. y= 2, .'. the roots of the equations are x=7, y=2, .'. a man earns $7 and a boy S2 per day. Verify by showing that these results satisfy the conditions of the given problem. 80 ALOEBEA Ex. 2. — For 3 1b. of tea and 2 1b. of sugar I pay $1-30, and for 5 lb. of tea and 4 lb. of sugar I pay $2 20. What is the price of one pound of each ? How does this problem differ from the preceding ? What change might we make in the first statement so that the number of pounds of sugar would be the same -as in the second statement ? Let X cents = the price of a lb. of tea, and y cents = the price of a lb. of sugar. Then 3a;+2j/=130, (1) and 5x + 42/ = 220. (2) Multiply the first equation by 2 and we get 6x + 42/ = 260, (3) 5x+42/ = 220. (2) Now solve (2) and (3) as in the preceding example and verify the •esults you get. Ex. 3.— Solve 3x+4y=39, (1) 4x+3i/=38. (2) Multiply (1) by 4 and (2) by 3 and we get 12a;+162/ = 156, 12x+ 9y=114. Complete the solution and verify. Ex. 4.— Solve 5x-2y^U, (1) 3x+4?/=42. (2) Multiply (1) by 2, 10x-42/ = 88. (3) To get rid of the term containing y, we must now add instead of subtract. When we do so 13a;=130, .-, x= 10. Substitute x= 10 in (1) and y= 3. 56. Elimination. In all of the preceding examples the object has been to get rid of one of the unknowns, so that we might have an equation with only one unknown. The process by which this is done is called elimination. SIMULTANEOUS EQUATIONS 81 Thus in Ex. 4 we eliminated the y. We might have ehminated the X equally well. Solve Ex. 4 by first eliminating the x. After performing the necessary multiplications, when do we add •ind when do we subtract to eliminate the unknown 1 EXE1ROISE3 41 Solve for x and y and verify 1-21 : 1. x+22/=8, x+ y=5. 4. 2x \-3y=25, 2x—3y^ 7. ^. 3x+5y=lS, 2x-\-3y=12. 10. x-{-y=i, x-y=3. 2. 3x+5y=13, 3x+2y= 7. 6. 5x—2y=lS, 2x- y= 7. 8. 5x—6y=3l, 6x-3y=33. 11. 3x+ iy= 5, 6x+12y=13. tk. 13. 3x— 4t/=16, ^ 14. 2x+5y= 0, lx+3y=62. yi^^ 16. 3x=2y+ 7, O*^ 2x='6y-l2. 19. 2x+l3y= 275, 14x-17t/=1385. 3x-'iy=23. 17. x^3y+20, y=2x-20. 3. Qx+5y=23, 3x+2y=n. 6, 5x+2i/=24, 2x+3y=U. »' 3x-2y=2A, 2x-3y=\\ 12. 3x+2^^24, — 2a;+3i/=10. 15, 2i/— 3a;=— 22, 2x+3i/=32 18. 3a;=2y, 2x-by=-33. 20. 22;+3i/= 5x— ^=17. 21. 4x-52/=10?/-14a;=-ia 22.* If 5x— 1/=8 and 5?/— x=20, find the values of x-\-y and x— j/, 23. If 2a:-5^-31=6(/-9a;+57=0, find the value of 19a;4-13y. 24. Solve x+3=4— 2i/, 7(x— l) + lli/=6. 25. If ax-\-by equals 39 when o is 3 and b is 4, and equals 13 when a is 5 and 6 is — 2, find x and y. 26. What values of x and y will make 16x— y and 4:X-\-2y each equal to 6 7 27. Solve 2(«->/)+3(x+?/)--3l, 3(2x-y)-f 5(a;-22/)=53, 82 ALOEBRA 57. Fractional Equations in two Unknowns. If the equations contain fractional coeffioients of x or y, the fractions may be removed by multiplication. Ex.— Solve ^x^^y^ 8, (1) ^a;+f?/=32. (2) Multiply (1) by 6, 3z+ 2y= 48. Multiply (2) by 4, z+10!/=128. Complete the solution and verify. EXERCISE 42 Solve and verify 1-20 : 1. iz+ly^S, 2. ix + y- 6, - i{a;+!/)=9. x-f y=l. x + |=14. J(x-2/)=4. 4. 6^5 5. iz + Zy=2, /•^ 8^3 - + ^ = 24. 9 2 x+iy=0. ?_y= 4 4 5 7. i^-y= 1, 8. ? + ^ = 41, 3 8 &. ^ + 92/= 91, |x+fy=26. 3x-4j/= 0. 9x + |=.67. 10. 16 ^ 24 11. x=y. 12. ix+ii/=6, f-^ = l. ,1 TO 9?/-llx-80. t/-J(x-j/)=7. 13. •3x+-5?/=-23, 14. -1x4- 32/= 2-6, 15. •05x+-03y=29, 6z^ oy=2-6. x-l-6y=10-2. •03x--04!/= 0- 16. x = ^ = ^. 17. ?_^ = f-^ = 3. 18. ^=^ = 6-y. 3 2 5376 25 * 19. x + | = t/ + | = 7. 20. |_f = 3x + 7y + 26 = 6. 10 SIMULTANEOUS EQUATIONS 83 21.* ^ + :^= 2i 22. z^-ky-l^y+Kx+y), 23. ar+|y=y-2, 24. 5(a;+2/)-7(a;-2/)=26, „„ x+l 3w— 5 a;— V 26. 8a;-7i/-12, 26. ^ - ^ - ^. x—2y 2x—y _ . 4^3 27. |?/-ix+24=f2/+^a;+ll=0. EXBRCISB 48 Solve, by using two unknowns, and verify : 1. The sum of two numbers is 40 and their difference is 12. Find the numbers. 2. The sum of two numbers is 19. The sum of 3 times the first and 4 times the second is 64. Find the numbers. 3. If 41b. of tea and 7 lb. of sugar cost $2-42, and 5 lb. of tea and 3 lb. of sugar' cost $2'68, find the cost of each per lb. 4. Find two numbers such that 7 times the first is greater than twice the second by 23, and 5 times the first and 3 times the second make 136. 5. If 5 horses and 6 cows cost $840, and 3 horses and 2 cows cost $440, find the cost of a horse. 6. If either 9 tables and 7 chairs, or 10 tables and 2 chairs, can be bought for $156, what is the cost of each ? 7. If 3 men and 4 women earn $164 in 4 days and 5 men and 2 women earn $135 in 3 days, find the daily wages of a man and of a woman. 8. Find two numbers such that \ of the first and \ of the second is 26, and \ of the first and \ of the second is 8. 9. Three bushels of wheat cost 20 cents more than 5 bushels ot com, and 2 bushels of wheat and 1 bushel of com cost $2 '30. What is the price of each per bushel ? 6 2 84 ALGEBRA 10. In 10 years a man will be twice as old as his son, but 8 years ago the man was 8 times as old as his son. Find their present ages. 11. If the sum of two numbers be added to 3 times their difference the result is 18 ; if twice the sum be added to their difference the result is 26. Find them. 12. A merchant sells 33 suits, some at $35 each- and the others at $25, and receives $945. How many did he sell at each price ? 13. Find two numbers such that 5% of the first is greater than 6% of the second by 3, and 7 % of the second is greater than 4 % of the first by 7-5. 14. If 3 algebras and 4 arithmetics cost $2-95, and 2 algebras and 3 arithmetics cost $2' 10, find the cost of 6 algebras and 2 arithmetics. 15. A bull's eye counts 5 and an inner 4. In 10 shots a marks- man scores 46 points, each shot being either a bull's eye or an inner. How many of each kind did he make ? 16. A classroom has 25 seats, some double and some single. If there is seating accommodation for 42 pupils, how many double seats are there ? 17. A man bought 8 cows and 50 sheep for $900. He sold the cows at a gain of 20% and the sheep at a gain of 10%, and received in all $1030. Find the cost of a cow ? 18. If 10 men and 8 boys receive $37, and 4 men receive $1 more than 6 bo3's, how much does each boy receive ? 19. A man bought 20 bushels of wheat and 15 bushels of corn for $36 and 15 bushels of wheat and 25 bushels of corn, at the same rate, for $32'50. How much did he pay per bushel for each ? 20. Find two numbers such that, if the first be increased by 8 it will be twice the second, and if the second be increased by 31 it will be three times the first. 21. A farmer bought 100 acres of land for $4220, part at $37 and the rest at $45 per acre. How many acres were there of each kind ? 22. Find two numbers such that 7 times the greater and 5 times the less together make 332, and 51 times their difference is 408. 23. The quotient is 20 when the sum of two numbers is divided by 3, and the quotient is 7 when their difference is divided by 2. Find the numbers. SIMULTANEOUS EQUATIONS 86 24. A grocer bought tea at 60c. a lb. and coSee at 30c., the total cost being S96, He sold the tea at 75c. a lb. and the coffee at 35c., and gained .$21. How many lb. of each did he buy ? 25. Three times the greater of two numbers exceeds twice the less by 90, and twice the greater together with three times the less is 255. Find the numbers. 26. The sum of two fractions whose denominators are 2 and 5 respectively is 2-9. If the numerators be interchanged the sum would be 4*1. Find the fractions. 27. Divide 142 into two parts so that when the larger part is divided by 17 and the other by 19 the sum of the quotients will be 8. 28. A farm was rented for S650, part of it at S6 and the rest at S8 per acre. If the rates had been interchanged the rental would have been S750. How many acres were in the farm ? 29. ^'s age 3 years ago was l^alf of fi's present age. In 7 years the sum of their ages will be 77 years. Find their present ages. 30. A man travelled 240 miles in 4 days, diminishing his rate each day by the same distance. The first two days he went 136 miles. How far did he go each day ? EXBRCISB 44 (Review of Chapter VII) 1. Solve 2x+3i/ = 38, 3x + 22/ = 37. 2. I fire 20 shots at a target. If a hit counts 5 and a miss counts — 2, how many hits did I make if my net score is 51 ? 3. Solve 7x — 22/=13, 2a; + 3y = 43. 4. The average marks of those who passed an examination was 65, and of those who failed was 25. If there were 1000 candidates in all and their average was 53, how many passed ? 5. Solve 2(a; — 2/) = 3(a;-4?/), 14(x + 2/)= ll(x + 8). 6. At an election A' a majority was 384, which weis ^'j of the whole number of votes. How many votes did A receive ? 7. Solve ^(a;+5)-5=§(2/ + 2), ^(y + 8)-3 = J(a;-3). 8. Divide $5600 into two parts, so that the income from one part at 3% may be equal to the income on the other part at 4%. 9. Solve f 4- ? = 3x - 7j/ - 37 = 0. 86 ALGEBRA 10. Two numbers differ by 11, and J of the larger is 1 more than t of the smaller. Find the numbers. 11. If px + qy is 74 when p = 5 and q—3, and is 76 when p = 6 and q = 2, find x and y. 12. If 3% of ^'s salary plus 4% of B's salary is $93, and 5% of ^'s plus 3% of B's is $111, find their salaries. 13. Solve 2l2/ + 20x=165, 77i/-30x=295. 14. Divide 100 into two parts so that | of the greater part exceeds J of the less by 2. 15. Solve 5x—2]/ = 7x + 2!/ = x+2/+ 11. 16. If 3 men and 4 boys earn $26, and 5 men and 2 boys earn $34, what would 7 men and 3 boys earn ? 17. Solve J(x+l)-J(!/ + 2) = 3, J(a; + 2) + i(]/ + 3) = 4. 18. If 3x — 4 = ax + 6 when x = 2 and when x=5, show that a = 3 and b=-4. 19. I bought a horse and carriage for $400. I sold the horse at a profit of 20% and the carriage at a loss of 4%, and on the whole transaction I gained 5%. What did each cost ? 20. Solve ^ - 27/ = 2x - ^ = 7. 2 * 2 21. A man pays a debt of $52 in $5 bills and $1 bills. If the number of bills is 24, how many are there of each ? 22. Solve 19x-21i/= 100, 21x- I9y= 140. 23. A's wages are half as high again as B's, but A spends twice as much as B. If ^ saves $5 and B $10 per week, what are the wages of each per week ? 24. If 23x+15y = 91, and y is 50% more than x, find x and y. 25. When a man was married his age was I more than his wife's age. His age 8 years afterwards was } more than his wife's age. How old was he when he was married ? 26. If 3(5x-2i/) = 2(3x4- 6t/), find x in terms of y. 27. A man has two farms rented at $5 per acre and the total rent is $1100. WTien the rent of the first is reduced 20% and the second is increased 20%, the total rent is $1120. How many acres are there in aach 7 SIMULTANEOUS EQUATIONS 87 28. If I + I = I + i^ = 9, find the value of | + | . 29. Seven years ago B was three times as old as A, but in 5 years he will be only twice as old. What are their present ages ? 31. Solve?2±5 = 6 + ?±-', !^« = 3 + ?±?. CHAPTEK Vlll TYPE PRODUCTS AND SIMPLE FACTORING 58. Factor. When a quantity is the product of two or more quantities, each of the latter is called a factor of the given quantity. Thus, the factors of 3hc are 3, b and c. The product of 6-f-c and a is ab-^ac, .'. the factors of oA + oc are a and 6 + c, or ab-\-ac = a{b + c). Siroilarly, ab — ac = a{b--c). When x-]-y-\-z is multiplied by a, the product ax-{-ay-\-az contains the factor a in each term. If we wish to factor ax-\-ay+az, we recog- nize that since a Ls a factor of each term, it °) °^-r°y+°^ x-\- y-{- z must be a factor of the whole expression. The remaining factor is the quotient found by dividing the expression by a. Then ax^ay-\-az=a{x+y+z). This is seen to be similar to the method in arithmetic. If we wish to factor 485, we see that 5 is a factor. How do we obtain the other factor ? Ex. — Factor 4a2— 6a6. Here we see that 2 and a are factors of each term and therefore 2o 18 a factor. On division the other factor is 2a — 3ft. .-. 4a*-6a6 = 2a(2a-3ft). Similarly, 36a; + 6cx = 3x( ) ■ ab--a*~a^^ a( ). The result of the factoring may be verified by multiplication and this may usually be done mentally. 81 10. 22/+4. 11. 13. ab+ac. 14, 16. mx-\-my—mz. 17, 19. 4x3 +6*2+ 2a;. 20 22. 2ax—4:ay-\-6az. 23, TYPE PRODUCTS AND SIMPLE FACTORING S» BXERCISE 45 Fill in the blanks in the following : 1. 4a:+6=:2( ). 2. 3a-9=3( ). 3. 5x-l0y=5{ ). 4. ax-\-3x=xi ). 5. bx—by=b{ ). 6, x^+x=x{ ). 7. 7p^—6p=p{ ). 8. 6y^^3y=3y( ). 9, 8a;3— 2a;2=2a;2{ ). Factor the following and verify : 6m- 12. 12, 3x2—15. am—bm. 15, ab-\-ac+a. x'^-lx. 18. 5a2-fiOa6. a^x-\-a'^y—a'^. 21. \5x^—\0xy. x^—3x^y+xyK 24, 4a6 + 6a262- 8a6c. 26, {x^y)a-\-{x-\-y)b. 26, r(a-6)+y(a-6). 27. 2a;(6-c)-2(6-c). 59. Definition. An algebraic expression containing only one term is called a monomial, one of two terms is called a binomial, one of three terms a trinomial, and one of more than three terms a multinomial or polynomial. Thus, 2x— 5 is a binomial and a* + 3a + 7 is a trinomial. 60. Product of two Binomials. The pupil should be able to write down mentally the product of two simple binomials like a; +2 and a; +3. x +3 What is the source of the first term (x*) in the product ? What is the source of the last term (6) ? What two ^ +^^ quantities were added to give the middle term (5x) ? +ox+d How were these two quantities obtained ? x*4-5x-i-6 In the product of x+1 and x+7, what would be the first term, the last term, the middle term ? What is the complete pioduct ? lu the product of x— 2 and x— 3, what is the first term, last term, middle term ? How does the product differ from the product of x + 2 and x+3 ? Ex. — Multiply x—5 by x+3. Why is the last term negative ? The middle term is the sum of + 3x and — 5x or — 2x. What is the complete product ? What is the middle term in the product of x+5 and x— 3 ' 90 ALGEBRA The middle term in every case is seeu to be the sum of the two cross products, each taken with the proper sign. r-f 5 x-Z State the pn 1. x+l x+2 I aducts 1 2_ 6. 10. 14. (x+2tj) 3XERCI af: x+ 5 x+11 SE {X- {m 46 (1 3. 7. 11. 16. -y)ix- +in){7 22. Oral) x-3 x-4: a+ b a+2b 8. 12. It (6- X- 5 «-12 5. J/-6 y+5 m-2 m+4 x-Sy x-2y 9. x+Ay x—3y 2/+5a; P- 6? P+Uq pq-r pq+r a-2 13. ab-l ab-3 xy-1 xy+1 18. 21. ax— 26y tx-362/ 17. (a+2)(( 20. (x-Sy) -4y). 19. n—5n). 22. q'.iv-q). 3K6-i). Remove the brackets, simplify and check : 23. 3(x+2)-f2(3x-l)-(x-3). 24. (x+l)(x+2)+(x+2)(x+3). 25. (2/+3)(2/-2) + (2/-5)(2/+4). 26. (x+l)2+(a:-l)(x+l)+(x+l)(x-2). 27. 2(TO+l)(TO+2) + 3(m-l)(m-2). 28. 4(x+3)(x+l)-(x+l)(x+12). 61. Factors of Trinomials. The product of two binomials, like those in the preceding exercise, is seen to be a trinomial. To find the factors of a trinomial we must reverse the process of multiplication. Ex. 1.— Factor a:2+6a:+8. Since the last term is positive, the last terms in the factors must have like signs, and since the middle term is positive, the signs must both be plus. ;, the factors axe of the form («+ K*+ )' TYPE PRODUCTS AND SIMPLE FACTORING 91 The last terms in the factors must be factors of 8, so they must be 1 and 8 or 2 and 4. x+1 x+2 x+8 a;+4 Which of these when multiplied will give the proper middle term T What are the factors of x* — 6x+8 ? The factors of x*— 9x+14 must be of the form (x— )(x— ). What are the factors ? Ex. 2.— Factor x^-2z-l5. Here the factors must be of the form (x— )(x-(- ), since —15 must be the product of two numbers differing in sign. The possible combinations are : x-15 x+ 1 X+I5 X- 1 x-5 x+3 x+5 x-3 Which of these sets of factors is the correct one ? In factoring a trinomial like x* — 8x+15, we require two factors of 15 whose algebraic sum is —8. They are evidently —5 and —3. .-. x»-8x+15 = (x-5)(x-3). In factoring x* — 4x— 21, we require two factors of —21 whose algebraic sum is —4, and they are evidently —7 and 3. .-. x*-4x-21 = (x-7)(x + 3). The pupil is advised to write the factors under each other, below the expression he is attempting to factor. Thus, X* — 6x— 16 X - 8 X +2 x»-6x-16 = (x-8)(x+2). x*+nxy-4:2y* X +14?/ X — 3y x* + Uxy-4:2y* = (x+Uy){x-3y). Factor: 1. x2+8x+7. 4. a^+22a+2\. 7. a^+3ab+2b^. 10. x2-5x+6. 13. x*—ixy+3y'. EXERCISE 47 (1-15. Oral) 2. x2+6x+5. 3. 5. x2_^8x+12. 6. 8. m^+lmn+lOnK 9. 11. x2— 7x+6. 12. 14. a^-Uab+28b^. 15. t/2+8y+15. 62+106+24. 2/2+40xt/+39x*. x2-12x+ll. m^— 7mn+12n«. 92 ALGEBRA 16. X2-X-20. 17. y^—y-30. 18. a^+a—30. 19. x2— 5x— 14. 20. m2-6m-40. 21. a:2_i0a;-24. 22. a^b^+8ab + 15. 23. x2y2_ii_j.j^_^30. 04, a;4_i0a;2+9. 25. a2+6a + 9. 26. x^ — 14x+49. 27. y*—\2y'^+36. Use factoring to simplify the following : -_ ^ a2^5a-4 a2+4o-5 _. m^— 5to+6 m^-7m+l2 ^o. -f— . ^y. — ^B a-ri a+5 m— 3 m— 4 (x''+3x+2)(x-5) 3x2--6x 2x3-4x2 x2-5x-f-4 x2-3x-10 ■ • 3x ^ 2x2 "^ x-1 * 32. What factor is common to (1) X2-X-30 and x2-2x-35 ? (2) a^+ab and a2+3a6+262 ? Find three factors of : 33. 2x2-10x4-12. 34. 3a2+3a-36. 35. x3-8x2+7x. 36. If the expression x2-L7ftx— 6 has two binomial factors with integral coefficients, what are all the possible values of wi ? 37. Is the expression x2— 3x— 10 factored when it is written in the formx(x-3)-10? 62. Square Root of a Monomial. When a number is the product of two equal factors, each factor is called a square root of the number. Thus, 16 = 4x4, therefore a square root of 16 is 4. But 16= —4x —4, therefore a square root of 16 is also —4. Similarly, the square root of 25 is -(-5 or —5, and the square root of 9a* is -j-3a or —3a. Thus it is seen that every number has two square roots differing onl}' in sign. It is customary to call the positive square root of a number the principal square root. 63. Radical Sign. The symbol -y/" , called the radical sign or root sign, is used to indicate the principal square root of a number. Thus, ^25 = 5, Va* = a, V9x*y*==ixy. T7PB PRODUCTS AND 81 MP LB FACTORING 93 When both the positive and negative square roots are considered, both signs must precede the radical sign. Thus, V'9 = 3 not -3; -V9= -3 not +3, but ±V9=±3, and is read " plus or minus the square root of 9 equals plus or minus 3." Thus, \/4+V9 = 2 + 3 = 5, but +\/i±V'9=±2±3=±5or ±1. If we represent the square root of 16 by x, then 0:^=16. To solve this equation, take the square root of each side, .". x=±4. We might have said ± x= ± 4, which includes the four statments : +a;=-f4, ~\-x——4:, — ^=+4, — x=— 4. If both terms of the last two be multiplied by — 1, the statements become the same as the first two, which are represented by a;=±4. We see then, that it is necessary to attach the double sign to the square root of only one side of the equation. Ex.— Solve (x+ 1)2=25. Take the square root of each side, .'. a;+l = ±5, a;= + 5— 1 = 5— I or — 5— 1, = 4 or —6. Show by substitution that each root satisfies the given equation. EXERCISE 48 (1-16. Oral) State the two sqiiare roots of 1. 36. 2. 81. 3. 121. 4. 2i. 5. 2/2. 6. 62c2. 7. 25a2 8. 64a;2j/2, 9. }a2. 10. Im^n^. 11. isP*- 12. 6^x2. Solve the following equations ; 13. a;2=9. 14. 3a:2=75. 15. x2=4o2. 16. x^=a%\ 17.* (x+2)2= =81. 18. {x-3)^ =49. 19. (a;-5)2=0. 94 ALOEBRA 20. If the area of a square is 100 square inches, find the length ot its side. 21. If r is the radius of a circle the area is nr^, where 7r=3^ approximately. If the area of a circle is 154 square inches, what is the radius, or what is the value of r, if S}r^—154: ? 22. Find the radius of a circle whose area is 616 sq.-in. 23. If r is the radius of a sphere the area of its surface is given by the formula, area =4:nr^. If the area of the surface of a sphere is 164 sq. in., what is the radius ? 64. Squares of Binomials. If we multiply x-]-y by z-\-y, the result, which will be the square of x-{-y, is x^-\-2xy-\-y^. The diagram shows a geometrical illus- tration of this identity. The first and last terms in x^-\-2xy-\-y'^ are the squares of the terms of X'\-y, and the middle term is twice the product of x and y. Therefore, the square of the sum of two numbers is equal to the sum of the squares of the numbers, increased by twice their product. Also {x—yy=x^—2xy-}-y^. Therefore, the square of the difference of two numbers is equal to the sum of the squares of the numbers decreased by twice their product. In the square of a sum all the terms are positive, and in the square of a difference the middle term is negative. Thus, (3a + 26)«=(3o)* + 2(3o){26) + (26)«, = 9a* + I2ab +46». (5x-3t/)' = (5x)»-2(5x)(3!/) + (32/)«, = 25x*— 30xy +9y*. (ix-42/») = (Jx)*-2(ix)(42/) + (42/)», = Jx* — 4xy -flGj/'. *' xy xy y' TYPE PRODUCTS AND SIMPLE FACTORING Sfi EXERCISE 49 (1-16, Oral) What are the squares of : 1. a+l. 2. y+2. 3. m— 1. 4. «— 4. 5. 2a+l. 6. l-3x, 7. p-g. 8. 2x+3. 9, 2a— 3. 10. TO— 2n. 11. 3x—2y. 12. 4x— 3(x. 13. J— a;. 14. 2!/— J. 15. 3a;— f. 16. — x— 2 Simplify : 17.* (x+l)2+(z-D2. 18. (a-6)2+(a+6)2. 19. (2x-|-l)2+(a;-2)2. 20. (a+6)2-(a-6)2. 21. (3TO-n)2— (2to+7i)2. 22. {3x+2y)^—{2z-3y)^. 23. (a:+l)2+(a;+2)2+(x+3)2. 24. [z-l)^+{x—2)^-{z-3)^. 25. 2{a+l)2+3(o-l)2-5(o-2)2. 26. Find the value of a^-f-fe^-j-c^ when a=x—y, b=z-\-y, c=x—2y. 27. Simplify {x+l)^+{x-2)^+{x-Z)^-3{x-^)^ 28. From the sum of the squares of a;-l-2, a;-|-3, a;-f-4, subtract the sum of the squares of z— 2, z— 3, a;— 4. 29. SimpUfy {2a-3b)^+{3a-{-2by-{2a+2b)K 30. If two numbers differ by 2, show that the difference of their squares is equal to twice their sum. 2 31. By how much does the square of x + - exceed the square of 2, ^ X I X 32. Show that the sum of the squares of three consecutive numbers is greater by two than three times the square of the middle number. 33. The square of 1234 is 1,522,756. Find the square of 1235. 34. The square of 2^=2x3+^=^6^; the square of 5| = 5x6+J= 30J, etc. In the same way find the squares of 6^, 8^, 20^. Prove that this method may be used to find the square of any number ending in J. (Let the number be n+J.) 65. Square Roots of Trinomials. Any trinomial which is of the form a^-\-2ah-\-b^ or a^—2ab-\rb^ is a perfect square. In order that a trinomial may be a square, the first and ALOEBRA last terms must each be a square and the middle term must; be twice the product of the quantities which were squared to produce the first and last terms. Thus, 9x* + 24x3/+ I6y* -.a square, because (1) 9x* is the square of 3x, (2) 16?/* is the square of iy, (3) 24Ty is twice the product of 3a; and 4y. -•. 9a;*+24xi/+16?/* = (3x + 42/)». .•. the square root of 9a;* + 24x1/4- 16?/* is 3x + 4?/. Is 4m*— 12mn+9n* a perfect square ? What is it the square of T What in its square root ? Similarly. 25x*- 10x+ l = (5x- 1)*, 36x* + 24x + 4 = ( )*, a*6*-6a6 + 9 = ( )*. Why is a*4-5a6-f-256* not a square ? Is it the square of o + 66 7 How would you change it so'that it would be a square ? The square root of a^-\-2ab-\-b^ is a-\-b, but — (a+6) or — a—b is also a square root, since . (-a—b)^=a^+2ab-\-b^. It is customary, however, in stating the square root of a trinomial to give onl}' that one which has its first term positive EXERCISE 50 (1-24. Oral) Express as squares : 1. a;2+2x2/+j/2. 2. y^'-2y+l. 4. 4a2+20a-l-25. 5. 9a2-24a-fl6. 7. a^b^-2ab+l. 8. l-6y+9y^. 10. a^b^c'^-2abc+l. 11. x^+x+J. What is the square root of : 13. 9a2+12a+4. 14. x^-^xy+^y^. 16. 4a262_20a6 + 25. 17. 'im^+2m+l. 19. 4-4a+a2. 20. 9—12x4-4x2. Supply the missing terms, so that the following will be perfeci squares : 22. a^ + - • ^ + b^- 23. x»-. . .+4y\ 24. x^^Gx . . . 26, 4m*—. +9 26. 9a*+lSa, . . 27= . . .— 4a;v-|-4t/» 3. 4xH4x+l. 6. 16x2-8x-|-l. 9. 9x2— 18x?/+9t/2. 12. y^-xy+lx^ 15. l-6x+9x2. 18. a2-14a6+4962. 21. 9x^—30xy+25y\ T7P10 PRODUCTS AND SIMPLE FACTORING 97 a +b a —b a*-\-ab -ab-b* a* -b* 28. If 16o — ma-f4 is a perfect square, what is the value of m ? Give two answers and verify eacL 29. What is the square root of 9a;^+6a;+l ? Check by putting a;=10. 30. Solve the equations and verify : (1) Vx^+2z-}-l + Vx^+10z+25=U. (2) 3Vx^—Ax+'L—2Vx'^+6x+9=—2. (3) V9x^+6x+l + V&+Ix+l + Vx'^—2x-\-l = 13. 31. Show that 2^02-60+9-^02— 4o+4=3V'a2-2a+l-V4o2+4a+l. 66. Product of the Sum and Difference. The product ot a-\-b and a—b is a^—b^, :. {a-\-b){a-b)=a^-b\ Here the two factors multiplied are the sum and difiference of the same two quantities a and b, and the product is the difference of the squares of a and b. Therefore, the product of the sum and difference of the same two quantities is equal to the difference of their squares. Thus, {x+y){x—y) = x*—y*. (2a+36)(2o-36) = (2a)»-(36)» = 4o«-96». (3a*-6)(3a*+6) = (3a»)*-6* = 9o«-6». (J + 3x)(J-3x) = (i)»-(3x)» = i-9x«. 67. Factors of the Difference of Two Squares. Since a^—b^~{a-{-b){a—b), the factors of the difference of two squares are the sum and the difference of the quantities squared. The diagram shows how this identity may be illustrated geometrically. Thus, 9a;*-252/*=(3x)»— (52/)«, which shows that it is the difference of the squares of 3a; and 5y. Therefore one factor of 9x'— 25^' is the sum of 3x and 5y, and the other is the differeuoe of 3j; and 5v- That is, 9j;»-252/» = {3x + 52/)(3x-5i/,. Similarly, ll6m»-9-(47n)» — 3» = (4m+3)(4m-3> a-b I -I I X I X I b b 98 ALGEBRA If we wish to factor Sx'^— 2y2^ we should recognize that 2 is a factor of each term. .-. 8x*-2i/» = 2(4x»-2/») = 2(2a;+y)(2cc-y). EXERCISE 51 (1-24, Oral) State the products of : 1. m-{-n, m—n. 2. p-q> p+q- 3. a+2,i 1-2. x—5, z-\-5. 5. 2a -4-1 , 2a- -1. 6. 3x-2, 3x+2. (2a— 3a;)(2a+3x). 8. (4x+52/)(4, x-5y). (a:+i)(x-i). 10. (x2- -2y){x' '■+2y). 12. (--1) (^^+l)- 1/2-4. 15. a2-462. 16. 4m2— n2. x2-i. 19. 9-x2. 20. 1-I6a262. a*-25. 23. a262-49. 24. 992-982. 9. 11. {5x+ab){5x-ab). State the factors of : 13. x^-l. 14. 17. 4p2— 9?2. 18. 21. 25—49x2. 22. Simplify : 25.* (a-2)(o+2)+/^-a-l)(2a+l). 26. (2a-36)(2a-|-36)-(a+6)(a-6). 27. 2(x-32/)(x+3y)+2(3i/-x)(3?/+x). 28. 2{p-q)^+3(p+q){p-q)-5{p + 2q){p-2q). 29. Find the product of x—a, x+a and x^-\-a^. 30. From the product of x— 1, x+l and x2 -f-1, subtract the product of X— 2, x-|-2 and x2-(-4. Find three factors of : 31 34 3x2-3t/2. 32. 5x2-20. 33. a^—a. mx2— TOa2. 35. 5-45392. 36. x*—y*. 37. -nR^—nr^. 38. a(x2- -l)+6(x2-l). 39. Why is the difference between the squares of any two consecutive numbers always equal to their sum ? TYPE PRODUCTS AND SIMPLE FACTORINO 99 40. Simplify {a^-b^){a^-5ab+6b^)-^{a'^-3ab+2b*). a;2 yi 3.2 j,2 3.2 iQ 3.2 g 41. Simplify — -] — and -• x—y x-\-y X— 4 x+3 42. Solve ?!i:l + ?!i:i^ = 10 ; 2(x-5)(a;+5)=15+(a;-l)(a;+l). x+l X— 3 68. Numerical Applications of Products and Factors. In this Chapter we have developed certain formulae concerning products and factors. (1) (a-fe)2 =a2-2aZ) + 62. (2) (a+6)2 =a2+2a6+62. (3) (a+&)(a-6) =a^-b^-. These formulae are true for all values of the letters involved. By substituting particular numbers for the letters we will see how some arithmetical operations might be simplified. (1) Since (a-6)' = a*-2a6 + 6*, 99«= (100- 1)«= 10000-200+ 1 = 10001 -200 = 9801. 37«= (40-3)*= 1600-240 + 9= 1609-240=1369. 998* = (1000-2)*= = = 89«= (90-1)*= = = (2) Since {a + b)* = a* + 2ab + b*, 92*= (90 + 2)*= 8100 + 360+4= 8464. 121* = (120+1)*= 14400 + 240+ 1 = 14641. 75*= (70 + 5)*= = (3) Since (a + 6)(o-6) = a*-6«, 92x88 = (90 + 2)(90-2) = 90*-2* = 8100- 4 = 8096. 65x75=(70-5)(70 + 5) = 70*-5* = 4900-25 = 4875. 27x23 = (25 + 2)(25-2)= = = 87x93 = ( )( )= = = (4) Since a*-6*=(a + 6)(a-6), 53*- 52* = (53 + 52)(53-52) = 105x 1 = 105. 41*- 31» = (41+31)(41-31)= 72x10=720. 727»-627* = ( )( )= = 67»- 33» = ( )( )= = H 2 100 ALOEBRA 69. Some (leometrical AppliCwtons. (1) If a is the length of t\u ^ide of the large square and b the side of the small square ,he area of the shaded portion IS evidently a^— 6^. If we wish to find the area of the snade" part when a a=77 and 6 = 23, we have a a»-6* = 77»-23» = (77 + 23)(77- 23)= 100x54 = 5400. If a = 225 and 6= 125, find the difference in the areas of the two squares. (2) The radius of the large circle is E and of the smail circle is r. The area of the large circle is '^fR^ and of the small one is ^-^-r^, .". the area of the shaded part is "^^{R^—r^). If iJ = 39 and r=31, find the area of the ring. The area = V(«»-r») = V(39*-31») = V(39-f-31){39- 31) = Vx 70x8= 1760. If i? = 89 and r = 82, show that the area of the ring is 3762. (3) In the right-angled triangle in the figure it '^ is shown in geometry that 62-|-c^=a2 or b^=a^—c^ or c^^=a^ — 6^. If a = 41 and c = 40, find the length of 6. 6« = a»-c» = 41»-40»=81xl=81, .-. 6=-v/81 = 9. If a = 61 and 6=11, show that c = 60. SXERCISE3 52 Use short methods in the following : 1. Find the squares of 98, 999, 119, 58, 799. 2. Find the products of 91 X 89, 61 x 59, 47 X 53, 203 X 197. 3. Find the values of 522-48^, 79^-782, 2152-2052, 7252-2752, 6732-5732. 4. If x2=62-c2, find X when 6 = 13, c=12 ; when 6=25, c=24. 5. If 7x2=642-572, find the value of x. TIPS PRODUCTS AND SIMPLE FACTORING 101 6. Find the difference of the areas of squares whose sides are ii and b for the following values : a= i 41 1 13 29 83 15m 2-85 &= 1 40 1 12 21 17 14m 215 1 7. Find the difference in the areas of circles whose radii are R and r for the following values ; B= 4 14 25 51 19o 3-25 -= 3 7 24 44 5a 2-35 Z}{R^-r^)=\ 1 ! 2. 4m-12n. 3. ax — bx. 5. x« + 4x+4. 6. a*-2a+l. 8. x»-3a;+2. 9. yt-y-nO. 11. 100p»-8l3». 12. a»-19a-20. 14. 300- 3x«. 15. {x + y)*-\. EXERCISE 53 (Review of Chapter VIII) Factor : 1. 3x+6y. 4. Bac — 36c. 7. 4a«-9x». 10. 2a»-I8. 13. 2a» + 6a+4. 16. a«-6«. 17. Solve x»= 100; x» = i; 9x« = 4; ox*= 1-25 18. Write down the squares of . 2x— 3, 5x-6, 4x-32/, a-J, 6c- J. 19. What are the square roots of: a*+6a + 9, p*~Sp+lQ, m>M«-10mn + 25, a*-o+i, 16x*-40xi/ + 25!/» ? 20. How much must be added to the middle terrr jf 4a*+3a+9 to tuake it the square of 2a + 3 '' 21. What middle term must be inserted in 9x* . . . -j-25y* to snake it a complete square ' Give twc answers. 102 ALGEBRA 22." Find three factors of x^ — x, 3x*— 12, a^-3a* + 2a. 23. Solve (z-3)* = 25; 4(a;-i)» = 9. 24. If a = 7rr», find r when a=12-56, 7r = 314. 25. Find the values of 997«, 875*- 7o«, 97x103, 81x81, 86x94, using algebraic methods. 26. Find four factors of 2x«-32, a«- 13a»-f36, '27n'- 18m and a\x^-y>)-b*(x*-y*). 27. Simplify (6+ l)* + (6- l)* + (c-f l)» + (c- 1)«. 28. Simplify {x + y){x-y) + {x+2y)(x-y) + {x + y){x-2y). 29. If a = 92 and 6 = 88, find the values of ab, a* — b*, a* + b*, using algebraic methods, 30. Simplify (a-26)(a + 26) + (a-46)(o + 46)-2{a-36)*. 31. What are ail the possible values of b, if a;*+6x+42 is the product of two factors with positive integral coefficients ? 32. Simplify ?^^%"-i^V^^- ^ ^ x—y x—2y x—Zy 33. If the square of 426 is 181476, find the squares of 427 and 425. CHAPTER IX SIMPLE APPLICATIONS OF FACTORING 70. Highest Common Factor. When a factor divides two or more expressions it is called a common factor of those expressions. Thus, 4 is a common factor of 8, 12 and 20, and a is a common factor of a*, 2a and 3a6. As in arithmetic, the highest common factor (H.C.F.) is the product of all the simple common factors. Thus, the simple common factors of 3a '6, 6ab* and 9abc are 3, a and b, and therefore the H.C.F. is 3o6. In the case of monomials the H.C.F. may be -m-itten down by inspection. Ex. 1.— Find the H.C.F. of 6m^n, 12mH^ and 9mH^. (1) The H.C.F. of 6, 12 and 9 is 3. (2) The highest power of m which is common is m*. (3) The highest power of n which is conamon is n. .•. the H.C.F. is 3xm*xn or 3m*n. If the expressions are not monomials they must be factored when possible, after which the H.C.F. may be written down by inspection. Ex. 2.— Find the H.C.F. of a^-{-ab, ab-\-b^, a^-\-3ab-\-2b^. a* + o6 = a(a + 6), ab + b* = b{a+b), o» + 3o6 + 26* = (a+6)(o+26}, „•. the H.C.F. = a+fc. 104 ALGEBRA E3XERCISE 64 (1-12. Oral) Find the H.C.F. of : 1. 3, 9, 12. 2. 16, 24, 40. 3. 2a, 46, 8c. 4. 3x, 6a;, 12a;. 5. 4ax, 6aa;, 2z2. 6. a26, a6-, a«6«. 7. 3a;2, 4x\ 5x*. 8. 5a2, 10o3, 15a. 9. llxY', 34x2;/, 51x3. 10. 2a,a^+ab. 11. 6a-2, 4a;2 + 2x. 12. (a+6)2, a2-62. 13.* 2a + 4ft. 3a +66. 14. a2_62, a6-62. 15. m^ — rj*. m^ — 2wn-|-n 8 16. x2+x?/, xy-l-j/2, {x+y)\ 17. mn + 2n, m^ + 3m^2. 18. a2-3a+2, a'—5a^6. 19. x^—9, a;"— 7x4- 12, x2 -4a; -1-3. 20. y^i-2y-3,y^+y-2. 21. a24-2a6-f fe2, 2a2-26' 2 22. x2-10x+25, 3x2-75. 23. 6a6+462, 6a24-4a6. 24. a^—2a^+a, a^+a^—2a. 25. If a-\-b is a common factor of a^-\-mab-\-b- and a^-\-nab-\-2b^, what are the values of m and n ? 26. The H.C.F of a^b and 06^ is ab. Find the greatest common measure of the numbers to which a^b and a6- are equal when a=2 and 6=4, and compare the result with the value of 06 when a = 2 and 6=4. 71. Algebraic Fractions. A fraction has the same meaning in algebra as it has in arithmetic. Thus, I means 3 of the 4 equal parts of a unit, or the quotient of 3-^4. Similarly, j- means a of the b equal parts of a unit, or the quotient of a-^b. The fraction - is read " a divided by b " or ' a over 6." 72. Changes in the Terms of a Fraction. As in arithmetic, both terms of a fraction may be multipUed or divided by the same quantity (zero being excepted) without altering the value of the fraction. SIMPLE APPLICATIONS OF FACTORING 105 Thus. ^=|=ij=|o = etc. „. ., , am OCX . Similarly, r = r- = i — = etO"» b be box a*b* _ a6» _ 6» _ 6 a*bc abc be c ' 73. Lowest Terms. A fraction is said to be in its lowest terms when its numerator and denominator have no common factor. If it is not in its lowest terms, it may be reduced by dividing both terms by all the common factors. EXAMPLBa. 18_18-f-6 3 2. 3. 4. 42 42^6 7- 15a»6 _ \5a*b^5ab _ 3o» 25a6» " 25ab*-^5ab ~ 66 * x*—y* ^ ( x+y){x—y) ^ x—y ^ x*-\-2xy-\-y* {x-\-y){,x-\-y) x+y x* + 5xy-{-iy* ^ (x+y)(x+4y) ^ x+y ■x*+3xy—4y* (x—y){x+4y) x—y The attention of the pupil is drawn to the fact that it is factors and not terms which are cancelled from the numerator and denominator, 7 + 2 Thus, in the fraction q-t-o ^6 cannot cancel the twos and say that the fraction is equal to |, for the value of the fraction is y\, which does not 7x2 equal J. But if the fraction is - — ^ we can now cancel th.- twos and the resulting fraction is J, Similarly, — = - after cancelling, or dividing by the common factor a. But is not equal to -. a+c c It is thus seen, that no cancelling can be done until both terms of the fraction are expressed as products. 108 ALGEBRA KXBRCISE 55 (6-21 Oral) Fill in the blanks in the following : ^15___30____6a:y ' 20~ 4 ~ ~12~4a~ _ ax _ _ oc _ _ —5a _ Zam bx b bm 3 a^ ^ g<P+g) b 6(m+n) Gfe"" 6a'a; _ a^x _ a*a; _ _ <* * 12a2x2 2x2 a'-6' ^ ( K ) ^ * a*-2a6-[-62 ( )a ^.j ' Reduce to lowest terms : 6. li. 7. ??. 8. ^. ». ?*. 21 6 25 6c 10. ?^. 11. 1^. 12. 1-^!^. 13. §^. 00 6x 15to» 5a6 14 ^"+^ 15 3a»+6a ^^ 4x ^^ o(x-3) 6 ' ' 12a ' ' 2x2-8x' ' 6(x-3) 18 ^(^-^) 19 (^-l)(^-2 ) 20 ^°^^~y) 21 °(°~^) ■ (a;-l)2' ■ (x-2)(x-3)" ' &a^ ' ' (a-b)^' 22*?!z:3 23 y'-y 24 ^'-3^+2 - * x^—x ' y^—2y+l ' x^— 4x+3 25 "t'+7m +12 2g a'-&' g? 3^'"%' TO«+4m ' ' o2-|-5a6+46* " ' 3x2+9xy+6t/«' 28. «;z:^. 2». ^^. 30. ?;^^ a*— o 4a*— 4 x'— x 74. Multiplication and Division of Fractions. The methods by which fractions aie multiplied and divided in algebra ^re the same as in arithmetic SIMPLE APPLICATIONS OF FACTORING iO", Examples. 3 5 ^ 3><_5 _ 15 10 7_^_10 7 15_25_ ^' 21 ^ 3 • 15 ~ 21 ^ 3 ^ 4 ~ 6 " *• a c_axc_ac b d~ bxd" hd ah xhi X* ab x*y a b xy a* a xi/ o* X* X a*+os6 cd-\-d* _ a{a+b) d(c-\-d) _ ad c*+cd ^ ab+b' ~~ c{c + d) ^ b{a + b) ~ be ' SXERCISE 56 2. 1x^-^1. 3. 15 7 14 5. — x |-- 6. xy be 25x2 ^ 15^ ^ • 141/2 • 7y • 11 4a+6fe^ lOx ^, 5x 2a+36 ^^ a2_3a_i_2 a2-7a+12 a'^-^a-\-Z Simplify 4 5 3 1. - X X - • 5 6 4 a ft c 4. T X X -• 6 c a a r. V. h~ d" 5x X— 1 1<>,* X -. 1 3x- -3 U > 13 X2- -1 v-^- -5x+6 x'-i- -4 X^- -4x+3 2a 96 36" 4c' 2a . 4a 36 ■ 36 6a 2 56"" M2a. x^+xy . xy+y' a*+o6 ■ ab + b^ -5a+6 a2-6a+5 a^-5a+4: - a^-b^ a2+2a6-862 ___ a^+Sab-i b'' ""' a2-3a6+262 ^ a^-2ab-3b^ ' o2-4a6+362 " 75. Lowest Common Multiple. A product is a multiple of any of its factors. Thus, Sxy is a multiple of 3, of x, of y, of 3x, of 3y, of xy, and a* is a mxiltiple of a, of a*, of a'. When an expression is a multiple of two or more expressions it is a common multiple of those expressions. Thus, I2a*b* La a common multiple of 2a' and 3a6*- 108 ALGEBRA ,- , ^ ■ ■ u ■ The lowest common multiple (L.C.M.)' of two or more expressions is the expression containing the smallest number of factors which is a multiple of each of the given expressions.'*; .C^ Ex. 1.— Find the L.C.M. of Qz-y, 9xy^ and I2xy^. The numerical coefficient of the L.C.M. is evidently the L.C.M. of 6, 9 and 12 or 36. The highest power of x in any of the given expressions is x* and of y is y^, so that the L.C.M. must contain the factors x* and y^. .-. the L.C.M. = 36 xx»X2/S = 36x*!/'. Ex. 2.— Find the L.C.M. of a^-b^ and a^-2ab+b^. a»-6» = {o-fe)(a + 6). a*~2ab + b* = {a-b)*. :. the L.C.M. =(a-6)«{a + 6). Why is {a-b)(a + b) or {a-b)*{a + b)* not the L.C.M. 7 EXERCISE 67 (1-9, Oral) Find the L.C.M. of : 1. 3, 4, 5. 2. 10, 15, 20. 3. 2a, 4a, 6o. 4. a, ab, a^. 5. x^, xy, y^. 6. 2ah, 3ac, 66c. 7. lOa^, 15a2, 5a. 8. 3a^ 2a^, 4a. 9. Ga^fc, 4ab^. 10.* a2, a'^^a. 11. 3x, 3x*+6x. 12. ab+ac, h^+hc. ^3. 2X-I-2, a:"-!. 14. x^+xy, {x^-y)^. 15. x^-\, x^-^x+2. 16. a^-ab,ab-b^. 17. a^-fe^, a2-2a6+62. 18. x^-x,x^-x. ^19. 2x, 4a;+4, 2x^-2. ^. y^-3y+2, y''-y-2, y^-\. 21. Show that the product of x^+x— 2 and x^— x— 6 is equal to the product of their H.C.F. and L.C.M. 76. Addition and Subtraction of Fractions. If we wish to add or subtract fractions we must reduce them to a common denominator. As in arithmetic, the lowest common denominator is the L.C.M. of the denominators. SIMPLE APPLICATIONS OF FACTORING 109 Examples. 35_2_9 1P_^_ 9+10-8 _ 11 4 ■''6 3 "12''" 12 12 ~ 12 "12* ('^ .01 ac ah ac+ab b c ~ be bc~ be ' 3 _ 4 5 _ ^ _ i^ , Soft _ 36«-4a*+6qfe ay—bx 5. x»+x2/ a^+y» a;(a;4-j/) y{x+y) xy(x+y) 2x 2 2« 2 2z-2( z— 2 ) x«— 4 ~ x+2 ~ (x+2)(a;+2) ~ z+2 " (x + 2)(x-2) 2x-2z + 4 (x+2)(x-2) (x+2)(x-2) BZERCISB 58 (1-8, Oral) Reduce to fractions with the lowest common denominator : 1. 2, 5. 2. •^, ±. 3. 1,1. 4. l.A. 3 9 4 46 a a^ 3x 2x ^ 3a 4a ^ m r? ,16 „ 2 36 O. -— , -— . b. — , - . 7, _, _. S. -, -— . 43 nm oc a a'^ ' 3a' 4a2' 2a3* ' 6' c' a* ' 6 ' 2 ' 36c ' 12 -?_ ^ A. 13 ^ ^ _L 14 °+^ g— 1 g+2 3t/2' 3xy' 2x' * 3aA' 26c' 4ac' * o ' 2a ' 3o " Perform the operations indicated : 15. ? + ?. 16. ^ + ^. 17. ^ + 5::^. 3^4 2^3 3^5 18. ^^_?. 19. -^ + J-. 20. ^ + ^%°^. 3 5 x-l^x+1 2 ^ 3 ^ 4 oi _1_ < J_ 22 * ^ - i^ 23 ^±^ + ^i:^ - ^+^ I' " ■ l-x"^l+x "3 6 ■ 4 "^ 2 8 ' 24 '^Z^-'^IZl + 'Lty. 26 -^ ^. 2 3^4 x-y x+y no ALGEBRA 26. -JL + JL_ 8^. 27. ^±^4-?!:? + a+4 a— 4 a*— 16 o x ox XX 9 a 28. -^__^. 29. 3x+6 2x+4 a*— aft a6— 6* 30. ——^ s + ^r-^ h-3— ^ ^- (Check when o = l.) a*+3a+2 a«+5a+6 o«+4a4-3 31- li-T + ^T 1 1 q!— 1 ' o*+3a+2 a«+a-2 77. Mixed Expressions. An expression which is partly integral and partly fractional is called a mixed expression. A mixed expression in algebra corresponds to a mixed number in arithmetic. Thus, 31 i3 a mixed number and a + - is a mixed expression. c Note that in a mixed number the sign of addition is omitted and 3r- means 3+?. But in algebra the sign must be inserted, as a - would mean ax- and not a -\ — . c c c 78. Reduction of a Mixed Expression to a Fraction. Since every integral quantity may be written as a fraction whose denominator is unity, it follows that the reduction of a mixed expression to a complete fraction is a problem in addition or subtraction. Examples. 2 15+2 17 c e c c a; 5y a; 5y-x a. o = — — -= y y y y ac a(6 + c) ac ab+ac—ae _ ab *• °~bTc^~b+c b + c~ b + c ' " b+e° SIMPLE APPLICATIONS OF FACTORING 111 79. Reduction of a Fraction to a Mixed Expression. To separate into two fractions we merely reverse the operation of addition. „ ab-{-bc ab , be , . be rhus, = = b -\ , a a a a ab — bd—c ab bd c , c and 6— =T- 6 -6 = °-'^-6' EXERCISE 50 (1-9, Oral) Reduce to complete fractions : 1. 2+|. 2. 1+^. 3. 3 + ?. 4. aH 5. x — ^- 6. 3 — -. ^4 3 c 7. x-^*. 8. 2x + ^. 9. a6-^. n X c 10. a — - 11. x-\ ^. 12. 2a H -. b+c x—y a—b 13. x+l + l. 14. a-6-^. 15. .-j/ + "'+^ 3z 2 — - » i ^_^y Separate into fractions in their lowest terms : 16 ^+^ 17 ?^±^. 18 5x-8j/ 4 ' ' ab ' ' 10a ' 19 6a2-36* 20 ^'^^"^ 21 ^iJi? 3o6 * ' 21a6 ' ' 6xy ' .„ 3»in— 4n _„ 6a6c— 96c-|-c' „. (a— 6)*+x ^^ • ^-^r * *.0» — • ^*. r • 2n 3oc a— EXERCISE 60 (Review of Chapter IX) 1. Define highest common factor and lowest common multiple. 2.* Find the H.C.F. and L.C.M. of 3a;- 6, 4a;- 8, 5x-10. 112 ALGEBRA 3. Find the H.C.F. and L.C.M. of x' + xy, xy + y* and x*y + Qry*. 4. Find *.he H.C.F. and L.C.M. of x»— 7x+10 and x' + 2x--8. Show that the product of these expressions is equal to the product of their H.C.F. and L.C.M. 6. Reduce to lowest terms : a*+ab X* 6a' — 9ab abx—bx* a* ' x* — xy' 8a6— 126*' acx — cx* /6. Multiply 5^. ;2' 2^' „ „. ,., 2c* 3bd 6cd /• Reduce to lowest terms : x*-2x a;» + 4x + 4 2g» -18 o*- 6« x»-5x+6' x» + 5x + 6' 3x» + 3x-r8* a*-2ab+b»' x~y x+y 2ax xy — y' J6. Simplify ,— „ *,-, X -i — K^~-^ X X , , • '/» T^- ^ o-x . o»— OX , x+2 x»— 4 0. Divide ,,„ by , — ^r^ and —- by ^ — |' o» + 2ax •' o*— 4x* x-j-l ' x*— 1 \/12. Fi ., x-3 , x + 4 , 2x-l , 8-4x Simplify 3 + ~4 ^^^ 3 ' g" ' Find the sum of x—y y—z z—x xy yz zx Zo T, u t 36 + 4a , 6-6c , ^ ^ a+6c /IB. From the sum of and ^t— subtract v— - • I ^ „. ... x-3 x-5 3x-2/ 32-21/ 4. Simplify ~K— — -= — and — ^ "^ 3x 5x xy yz a*--b* 15. Express ^r^- as the difference of two frMtions in their 5»_c* c* — a* lowest terms. Do the same with , , , and , . and find the sum o'c c a of the three results. V — 4 1 V — 5 16. By how much does - — exceed -= — 7 ' 4y 5j/ 11 2a 17. Find the sum of — r-., . and , — j-,- a-\-o a — a* — o" SIMPLE APPLICATIONS OF FACTORING 113 3. 3 2;* 5a; +6 18. By wliat must — _. be multiplied to give -^ — n~~r'ity b,b the product ? «+4 . . x*4-a;— 12 19. Find tne quotient when — -j is divided by -j — - • 2U. Solve (o-6)x = (o»-6*)(o + 6). 21. Find the difference between a , b , c , X , X , X + z and h r a—x b—x c—x a — x b — x c—x by first subtracting from , etc. ' a — x a — x 22. Find the missing term in the following identity : x*--5x+6 x* + 5x. . . _ a;»+2a;-8 a!»-3x-4 ^ a;»-9 ~ x*-x-12' CHAPTER X REVIEW OF THE SIMPLE RULES 80. In this Chapter will be found such exercises as will furnish a review of the elementary rules. In it is also included matter which it was not thought advisable to present to beginners in the subject of algebra. 81. Use of Brackets. We have already seen that (1) a-^{b-\-c)=a-\-b-\-c, (2) a-{-{b—c)^a-\-b—c, (3) a—{b-\-c)=a—b—c, (4) a—{b — c)=a—b-\-c. That is, when brackets are preceded by the negative sign, as in (3) and (4), the brackets may be removed if the signs of all terms within the brackets be changed ; but when they are preceded by the positive sign, as in (1) and (2), no change is made in the signs when the brackets are removed. In (3) the sign of b in a—{b-\-c) is positive as the expression might be written a— ( + 6 + c). When the brackets are removed we follow the rule and change -\-b into —b. Sometimes we find more than one pair of brackets in the same expression. Ex. 1.— SimpHfy a-(3a-26) + (5a-4&). Following the rule, this expression becomes a— 3a + 26 + 5a — 46 or 3a — 26. When one pair of brackets appears within another, it is better to remove the brackets singly, and the pupil is advised to remove the inner brackets first. REVIEW OF THE SIMPLE RULES 116 Ex. 2.— SimpHfy 4:X-{2x-i3+x)}. Removing the inner brackets, we get 4x-{2x-3-x]. Removing the remaining brackets, we get 4x-2x+3 + x or 3x + 3. Ex. 3. — Simplify 3a— [a+6— {a— 6— c— (a+&— c)}]. The expression = 3a — [a+b—{a — b — c — a—b-\-c}], = 3a — [a+b—a-\-b + c + a+b — c], = 3a—a — b-{-a — b — c — a — b-{-c, = 2a-36. After removing the first pair of brackets, the quantity a—b—c—a—b+c might have been changed into the simple form —26. Work the problem again, simplifying at each step. When brackets are used to indicate multiplication, the multiplication must be performed if the brackets are removed. Ex. 4.— Simplify ^x—3{x—2y)-\-2z^^y. The expression =ix — (3i —Gy) -\-2x—^y, = 4x - 3x +62/ +2x - 4 y, = 3x+22/. Note. — When the pupil has had some practice he should be able to remove the brackets and perform the multiplication in a single step. EXERCISB 61 (1-e, Oral) Remove the brackets from : 1. (a-fe)+(c-rf). 2. (a-h)-{c-d). 3. -(a-6)+(c-d). 4. -{a-h)-{c-d). 6. a—{h—c)—{d—e). 6. a—{—b)-{—c). 7. a+{6+(c-d)}. 8. a+\b-{c-d)\. 9. a-{h+{c-d)]. 10. a-{b-{c-d)\. 11. a-{-h-{c-d)\. 12. -[a-\h-{c-d)\]. I 2 116 ALQEBhA Simplify : 13. 4a-26-(2a-2fc). 14. 2(7a;-3t/)-3(2a;— 3j/) 15. 3(a-6 + c)-2(a-^6-c). 16. 2a- :3a4-2(a— 26);-. 17. 3(a+6-c)-2(a-6+c)+5(6-c+a). 18.* 15a;-:4-[3-5x-(3x-7)];. ^. Add 3(a+6)— 5(/)4-?), — 2(a+6)+(p+?) and 4(p+g). 20. Add 1+z— 2/, 1—x—y and 1— z+y- 21. Add 3x-2(y-z), 3y-2{z-x), 3z-2{z-y). Remove the brackets and express in descending powers of x : . 22. 3{5x-3+2z2)_2(x2-5 + 3x)-3(4-5x-6x2). 23. 2x(3x-2)-5(x-3)+6x(x-l)-2(x2-5x). 24. ^(4x-3)-J(6-x2)+|(x2+8x-12). Solve for X and verify : 25. 4(x-3)-7(x— 4)=6-x. ^JtQ. 5x-[8x-3;i6-6x-(4-5x);]==6. ^7. 3(2x-7)-(x-14)-2{5x+17) = 6(5-8x) + 21z+149. 28. i(27-2x)==|-^(7x-54). ^9. Simplify a— [56— {a— (3c-36)+2c-(a-26-c)n. 2(5 S' lif ' S{a-b+c)+2{b-c+a)-{c-a+b) 1/ ' '"^P ^ 5{a-2b+c)-2{b-3c+2a)-{Uc-2a-nb)' 31. Solve (7jx-2J)-[4i-|(3J-5x)]=18J. 82. Insertion of Quantities in Brackets. The trinomial a—b-\-c may be changed into a binomial by combining two of its terms into a single term. This may be done in a number of ways. Thus. a-6 + c = (a-6) + c = (a + c)-6 = a — (6 — c) = a-t-(c — 6). Remove the brackets mentally and see that each of these is equal to o— 6+c. REVIEW OF THE SIMPLE RULES 117 Ex, 1. — Express a — h-\-c—d as a binomial. As we have seen, this may be done in many ways as a — (6 — c + rf), (a-6) + (c-d), (a+c)-(6 + rf), (o-d)-(6-c), c-(6 + d-a). Note. — The pupil must exercise particular care when dealing with brackets which are preceded by the negative sign. The signs of all terms inserted in such brackets must, of course, be changed. He should, in every ^case, remove the brackets mentally to test the accuracy of the work. Ex. 2. — Express a-\-h—c as a binomial by combining the last two terms within brackets, preceded by the negative sign. a-\-b — c = a—{ — b-\-c), = o— (c — 6). Either form is correct, but it is usual to make the first term within the brackets positive, so that the second form is preferable. 83. Collecting CoefiBcients. Brackets are frequently used for the purpose of collecting the coefficients of particular letters in an expression. Thus, ax-\-by — cx — dy = x{a — c)-\-y(b — d), and mx—ny-\-px-\-qy = x{m-\-p)—y(n — q), =x{m+p)-\-y{q—n). Verify these by removing the brackets, BXERCISB 62 1. Express 3a— 26+ 4c as a binomial in three diff^ent ways and verify in each case. 2. Express p—q—r-\-s as a trinomial in four different ways and p^erify. 3. Express x—y—z—k as a binomial in four different ways and verify. Collect the coeflBcients of x and y : 4. ax — by — ex — dy. 6. mx — ny — px-{-qy—ax-\-by. 6. a(x-j/)+6(2j/-3x)H-c(5x+2t/). 7. x(c— 6)+y(6— c)— d(x+y). 8. 2ax— 36y— lOx— 5j?+66x— 4oy. »o {a—%)v~(^—h)x-\-^y-\-2ax—{Zx-\-h^), 118 AWE BRA 10. Enclose a- b—c—d—e-^f in alphabetical order in brackets, with two terms in eich ; with throe tcrrriH in ea<;h. Arrange in descending pxiwcrs of x : 11. a<z2+4-3z)-6(3a:-5x2)-c{l-4;c). 12. (ix'^-hx+c-(2jjx^-'iqx+r)—{ldx^^-'icx+f). 84. Multiplication with Detached Coefficients, The method of multiplying two bino.aiaLs has already been shown in Chapter V. The same method is followed when the factorf? are not binomials. (3) Check x=\ + 2 -1 The second method is called multiplication with detached coefficients. The proce.Hscs in the two methods are the same, with the exception that the letters are omitted in the second method and the coefficients only are used. When the second method is used the first coefficient in the yjroduct must h>e the coefficient of the product of x^ and X, that is, of x^. The next must be the coefficient of x^ and the next of x, as the product will evidently be in descending powers of x, as both factors multiplied are so written. In {'4) the check is shown as explained in art. 42. Ex. 2.— Multiply ^x^~lx-\-2 by a;2-2x+3. Here the term containing x* in the first 3 + 0—7+ 2 exproHsion is rniBsing and a zero is Hup- 1—2 + 3 plie<^J irj* order to bring coefficientH of like " ' ^ '14-0 7-t-2 powers of X under each other in the partial "^ ' "t Ex. 1.— Multiply ar»-3x+4 X -2 X2- 8 -8 -3x+4 by x-2. (2) 1-3 + 4 1-2 x»-3x»+ 4x -2x*+ &x- 1-3+ 4 -2+ 6-8 x»-5x»+10x- 1 .5flO-8 -6-0+14- 4 + 9+ 0-21+1 products. The first term in tlie product is 'Ax''. Write down the complete product and 3 — 6 + 2+ 16 — 25 + ft che<^:k the work. HE VIEW OF THE SIMPLE RULES 119 Kx. 3. — Find the coefficient of .(•'' in the product of 5.1:3— 6.1-2 +3.r- 2 and .r^— 2.r2 -3.r~f--t. Horo the oomplete product )s not roqiiirod. but only (lio torin which contains ;c*. The partial products wliicli will contain x' arc evidently those which we obtain by multiplying —2 by .»', 3.v. by — 2.r*, — tU^ by — 3x and 5x* by 4. ,-, thp coefficient of x* = - 2 - 6 f 1 8 + 20 -= 30. Ex. 4. — Multiply ax~-\-b.v-{-c by 7nx—n. ax* -\-bx +0 Here the multiplication is ,,,^ performed in the usual way. In adding the partial [)ro- (n/i.r' | biiix* ~\-cin.v ducts, the coetVicients of tiie aii.r* biix —en powers of .v are collected. — -~— amx^ -\- {bm — an)x* + (cw» — bn )x — ciu BXBROISfil 68 Multiply and check : 1. a;2-3a;+2, «-2. 2. 2a;»-5.c-3. .'^.)-2. 3. x»-a;+l,a;+l. 4. a^-\-ab+b\ a h. 5. x^-x+l, x^+xi-l. 6. a'-6a»-2. aM «-l 7. 3.»:2-2a;-r>, .r2-f-;i--3. 8. 'Za^-^rnib \ :W\ 2a' \ [wh-'Ah^ 0. a \-b-c,a-b-\-c. 10. a;-''| 2.r-' | 4.r | S. .r2-4.r+4. 11. b^-b+l,b^+b+l,b*-b^-\-l. 12. x'—xy-j-y^-'rx-^-y-^-l, .r-fj/— 1. Use detached ooefVioionts to multiply ; chcok the results r 13. 3x3-4xa+7x-3 by a;2-2.r- 1. Ai. r>a*-6<i3-2(i'»-« I 2 by 2f/- :hi |-2. 16. 4x'-5a;-2 by 4j;a-3.c-l. 16. (x2-a;-2)(2x«-a- l)(3.r-2). Simplify : 17.* (a;-l)(x-2) + (x-2){j:-3)+(x-3)(x-l). 120 AWEBRA 18. (a+x)(6-c)+(6+a;)(c-a)+(c+z)(o-b). 19. (a+6)(c+d)-(a-6)(c-(f). 20. (o+6— c)(a— 6) + (6+c— a)(6-c) + (c+a-6)(c— a>, 21. (x+l)(z4-2)(«+3)-(a;-l)(x-2)(x-3). Find the product of : 22. (l_a;)(l+a;)(l+a;2)(l+a;*). 23. (x-l)(ar-2)(a;-3)(a;-4). 24. (x-l)(a;-3)(a;+l)(a;+3). 25. (a-l)(a2+a+l)(a3+l). Find the eoeffieient ofa;2 in the product of : 26. 3a:2_5x+ll and 5x2+6x-4. 27. a;3+4x2— 5x+2 and x^— 2x-3. 28. 3x2-12x+15 and 2x2-7x-38. 29. Multiply 1+x+x^+x^ by l4-2x+3x^4-4x', retaining no powers higher than the third. 30. Add together (x-l)(x+2), (x+2)(x-3), (x+3)(x+4)(x-l). (x+4)(x2-2x+3) and 7-x2+3x. Check by putting x=2. 3*: Multiply 7x3— 5x2y-x?/2-f 6?/' by 4x2+3x2/— 22/2. 32. Show that the expression x(x+l)(x+2)(x+3) + l is equal to ^ (x»+3x+l)2. 33. Find the first four terms only in the product of: 2 + 3x+4x2+5x3 and l-2x+3x2— 4x3. 34. Find the coeflBcient of x* in the product of : l+4x+7x2-j-10x3+13x* and l+5x+9x2+13x'+17a;«. X Solve and verify : (x-2)(t-4)(x-6)(x-10) = (x-1){x-5)(x-7)(x-9). 36. Multiply ax^-\-bx-\-c by bx^—cx-\-d. Collect the coefficieni^ of X and write in descending powers. 37. Multiply px^—qx-\-r by px-\-q, and {a—\)x^-\-ax—\ by ax+1. 38. Simplify (ay^-hy+c){ay+h)+{ay^^-hy-c){ay-b). ^^i^%. Subtract the product of x2-|-x(p+l) — 1 and x— 2p from the product of X*— x(j3— 1)+2 and x-f-j>. REVIEW OF THE SIMPLE RULES 121 40. Point out two obvious errors in each of the followin|^tatement8 : (i) ah{a^h){a^-]^b^) = a*b^a^b-\-a^h^~ab*. (ii) (2x+3?/)'=6x=' + 36xy-54a;2/2+27»/3. (iii) x^—%x^y—^xy^+2y^={x—2y){x^-Ax-\-y% 41. Use the formula (a+l)(6+l)=a6+(a+6) + l to find the product of 2146 and 3526, being given that the product of 2145 and 3525 is 7,561,125. 85. Division by a Compound Quantity. The method of dividing by a monomial has already been shown in Chapter V. The method of dividing by a quantity containing two or more terms is in many ways similar to long division in arithmetic. Divide 672 by 32. (1) (2) 32)672(21 3. 10+2)6. 10* + 7. 10 + 2(2. 10+1 64 6ol0*+4.10 32 3. 10+2 32 3.10+2 In (2) the divisor is expressed in the equivalent form, 3 . 10+2 and the dividend 6 . 102+ 7 . 10+2. ^ If we substitute x for 10 the problem would be : Divide Qx^-{-lx-\-2 by 3x+2. The method here is so similar to the 3a; + 2)6x* + 7x + 2(2a;+l method in arithmetic, that little explana- 6x* + 4a; iion is necessary. The first term in the quotient is obtained by dividing 3x into ix-\-Z 6a;». The product of 3a; +2 by 2x is then ^x+2 subtracted from the dividend and the remainder is 3x+2. The last term of the quotient is obtained by dividing the first term of the remainder (3a;) by the first term of the divisor (3x), In more compUcated examples the method is precisely the same as hers. The division is continued until there is J 122 J ALGEBRA no T-pmoinlbj^nr until a remainder is found which is of lower nan^The degree tljan^he divisor 86. * Verifying Division. The work may be verified as in arithmetic, by multiplication. It is simpler, however, to test by substituting a particular number for each letter involved. Thus, in the preceding problem if we let x=l, the divisor is 5, the dividend is 15 and the quotient is 3, which shows that the result is very likely correct. If on substituting particular values for the letters involved, the divisor becomes zero, other values should be selected. 87. Division with Detached Coefficients. As in multiplication the method of detached coefficients may be used. Ex.— Divide \4.x^-x^-2^x^-\-\2 by 7a;2+3a;-6. ^ 74.3- 6)14- 1-29 + 0+ 12(2- 1-2 14 + 6-12 Check x=l -7-17 + 4)-4(-l -7- 3 + 6 -14-6+12 -14-6+12 Here the first term in the quotient is 2x*, since 14a;*-f-7x* = 2a;*. The complete quotient is 2x'' — x — 2. Divide also by the usual method. BXERCISB 64 (1-6. Oral) State the quotients in the following divisions : a;2+3a;+2 „ a2-3a+2 ■ „ a^ 1. x+1 ' a—\ ' a—b a;2-4 , a2+2afe+62 _ 3a;2-5x+2 5. ' x+2 * a+6 ■ 3x-2 Divide and verify : 7. 6a;2+x-15by2a;-3. 8. 6x^+xy-l2y^hy3x-Ay. 9. 5x^—Zlxy+&y^hyx—&y. 10. 9a^-\-6ab—35b^hySa+lb. REVIEW OF THE SIMPLE RULES 123 11. 7x3+96a;2-28xby7a;-2. H^. lOOx^-lSx^-^x by 25a;-|-3. yt3. 3+7x— 6x2by3— 2a;. t^. 6a2+35— 31a by 2a— 7. ^^. x^+13x^+54:x+'12hy x+G. ^. 2a3+7a2+5a+100 by a+5. 17. x'^+Sx^y+^xy^+y^hy x+y. 18. —x^+ix^y—3xy^+y^hyx—y 19. 16to3— 46wi2+39m— 9 by 8m— 3. , 20. 6x3— 29x22/+18x2/H352/3 by 2x-7y. ^ j^^tf a*+a3+4a2+3a+9 by a^- a+3. » 22. x<— x3— 6x2+15x— 9by x2+2x— 3. t^. 5x*-4x'+3x2+22x+55 by Sx^+llx+lL 1^. 2x3-8x+x*-)- 12-7x2 by a.2_^2-3x. ^. 30-12x2+x*-xby X-54-X2. Use detached coefficients to divide : 26. x3— 3x2+3x— 1 byx2— 2x+l. 27. 6x*-x3-llx2-10x-2by 2x2-3x-l. 28. oS-5o3+7a2+6a+l by a2+3a+l. 29. 4x2+9+x«+3x+x3 by 2x+3+x2. •» 30.* (x2-x-2)(2x2+a;-l) by 2x2-5x+2. 31. ^Ox^+llx*-2s:^-\lx^-x-\^ by 2x''+x-l. 32. Divide a^— 1 by a— 1 and o'+l by a+1. x^-{-y^ x^—y^ 33. Simplify x+y x—y 34. Simplify ^!l±,+ "'+^ 02+0+l o2— O+l __ Q , 6x2+x-2 3x2+8x-3 ,, 3o. Solve — — -! — -J — - — =11. 2x-l 3x— 1 i§6j If x(3a2-a + l)+2=-3a3-7a2+3a, find x. 37. The dividend is a^+ea^+Ga^- 9o4-2, the quotient is a'+3o— 1. Find the divisor. 38. Divide x8+x*y*+^ by x^—x^y^-\-y* and divide the quotient by x^—xy-\-y^. 194 ALOSBRA o- 11/ abx^—x(ac+b^)-\-b€ , acx^-\-x(nd+bc)+bd ^ -^ bx—c cx+d Without removing the brackets divide ; 40. ax^+{b+ac)x+bc by ax+b. 41. x^+{2p-l)x+p{p—l) by x+p. 42. a^x^ — 2abx+b^—c'^ by ax—{b—c). 43. aV+(2a'+a)y'+(a'+2a)2/+(a+l) by ai/2+ot/+l. 88. Inexact Division. As m arithmetic, the divisor may not divide evenly into the dividend, and so there may be a remainder. Thus, 34-^5 gives a quotient 6 and a remainder 4, /. 3ji = 6+4 or 34 = 6. 5 + 4. Similarly, when a*-r3a + 5 is divided by a+1, the quotient is o+2 and the remainder is 3. a-t-1 o+l or a» + 3a + 5 = (a+l)(a + 2) + 3. That is, dividend=divisorxquotient+remainder. Ex. — Express -j as a mixed expression. Here the quotient ia 1— a;)l +x*{l+x + x* + x* l + x+x*+x' ^-' sind the remainder is 2x*. +x __ = ! + , + .. + .3+^_^ + x* Divide 1— x^by 1 + x and show that +x* — x' = 1 — X 4- X* — x» + X* — , ,- • +^ +* 1+x 1+x In such cases the division may, of course, be continued to any number of terms. + x»— x« 2s« MVIEW OF THE SIMPLE RULES i2« BXEROISB 66 Find the remainder on dividing ; 1.* a:2-10a;+25 by a;-7. 2. a2+20a+70 by a+6. 3. a:3_4^2_^53._^20by X— 1. 4. y^—7y^+8y-\ hy y^—y+l_ 6. x^+y^ by x— i/, 6. x^—y^ by x+?/. Express as mixed quantities : 7 ^±^ 8 ^±^^ 9 ^°~^^ 10 Q^^+7a:-3 a;4-l ' a— fe ' a+b ' x-\-2 Find four terms in the quotient of : 11. l-^(l-x). 12. l^(l+x). 13, 1±^. 14. l+a+2a' 1— X 1— a+a^ 15. When the dividend is a^— 3a + 7, the quotient is a and the I'emainder is 7. Find the divisor. 16. Divide x^— 5x+a by x— 2 and determine for what value of a the division will be exact. 17. If x^— mx+12 is divisible by x— 3, what must the quotient bfe \nd what is the value of m ? 18. By division show that — ' ^a+b + -; _L^=a-64-— — . a~o a—o a+o a-]-o BXERCISB 66 (Review of Chapter X) 1. Add x^-2ax*+a*x+a^, 3x» + 3ax*, 2o»— ox*— x». 2. Add ^a — }b,^a- lb, la+i'o. 3. Subtract 8a + 36 -5c from Ua- 26 + 5c— 3d. 4. Subtract — 3a + 46 — c from zero. 5. Subtract |a— ^6 + Jc from fa+§6 — |c. 6. How much must be added to Jx— iy+^z to produce x—y+z\ 7. Subtract the sumi of 3a + 26, 26 — 3c and 3c — a from the sum of a— 6, 6— c and c — a. 8. Simplify o-(36-4c)-(6+c-o)-2(a—c). im ALGEBRA 9. Subtract 4x* — 3x' — a;-|-2 from 7x' — 6x* + 2x— 1 and check by substituting 2 for x. 10. Find the value of o' + fe' + c' — 3a6c when a=2, fc=3, c=— 5. 11, Simplify {a + b-c)-{b + c-a)-{c + a-b)-{a+b+c). ,.,„,-*2T Multiply l-4x-10x' by l-6x+3x*. 13. Find the product of x+1, x+ 2 and X— 3. %4. Divide the product of x + 2, 2x-3, 3x-2 by 3x« + 4x— 4 and check when x= 1. IfiT* Multiply a*-'rb*-\-c* — ab — bc — ca by a + b + c. 16. Find the product of a- 2, a+ 2, a* + 4. o*+ 16. - 17. Divide x«+64 by x» + 4x+8. , 18. Divide x* + x* — 24x* — 35x + 57 by x* + 2x-3, using detached 'coefficients and verify by multiplication using the same method-. 19. Find the coefficient of x' in the product of 3x' — 2x* + 7x — 2 and 2x» + 5x»+llx+4. ' 20. The expression 44x* — 83x' — 74x*+89x + 56 is the product of ^two expressions of which 4x' — 5x— 7 is one. Find the other. , v21 Divide x* + 4x' + 6x* + 4x4- 1 by x* + 2x+l and check by BUJMtituting x= 10. 22. Subtract ax* + 6x + c from cx* + dx4-/. collecting the coefficients of powers of x in the result. y 23. Find the remainder on dividing x*+6 by x*— 1. 24. Show that (a — 6)(x — a — 6) + (6 — c)(x — 6 — c) + (c — a)(x — c — a) = 0. 25. Show that (a-6)(6-c)(c-a) = a{6*-c*) + fc(c«-a*)+c(o»-6*). 26. If a = x* + 2xy + 2y^, b = x*-2xy + 2y^, c = x*-y\ find the value of ab — c. 27. Subtract 2x-3{y + 2z) from 32/-(Bz-3x). 28. If « = a + 6 + c, find in terms of a, b, c the value of a(3 — a) + b{s — b)-\-c{s — c). 29. Arrange in descending powers of x, c{ax — b)—x(a — b)-\-bx(x* — cx). 30. When o = 5, find the value of 2a_i3a-(464-2oU + 5a-(46— o).. RE'9'fEW OF THE SIMPLE RULES 127 31. What quantity when divided by x* — 2a;+3 gives x* + 2x— 3 as quotient and 9 as remainder "> ^. If a = a;*-3x + 2, 6 = 3x»-10x+8, c = 4x*-9a;+2, find the value of (a+26-c)-^(x-2). 33. Arrange in descending order of magnitude and find the average of: 30, -15, 27, 0, 3, -10, -2, 6, -8. 34. What number must be added to 5x*— 13x'-|-2a;— 1 so that the sum may be divisible by x— 2 ? 35. Find the coefficient of x* when l+a; + x* is divided by 1 — x — x'. y^. Divide 3p*-7p(l-j3»)-(2+p») by (3j9+l)(p+l). 37. If X* — 9x + c is divisible by x-|-4, find c. 38. Find the sum of the coefficients in the square of 2x* — x— 3. 39. Find the product of x-\-a, x + 6, x + c. Collect the coefficients of the powers of x in the product. From the result, write down the product of x4- 1, a;-|-2, x + 3 and of x— 1, x — 3, x + 4. JLii. Divide x«-2x'+l by x*-2x+l. >1. When a = 3, h = 2\, c = 2, find the value of —-+ \/7a6(2c«-o6) - (2o-36)». ^ /42. Prove that (I+x)»(l+y>)-(I+x*)(l+2/)« = 2(x-2/)(l-xi/). 43. If p = X — - and q = x* j, show that j)*(p* + 4) = g*. j^'^. Divide a' + 6' + c' — 3a6c by a + 6 + c. What are the factors of o'4-6» + c*— 3a6c ? Compare with Ex. 15. i^^ft5. Multiply x* + fex4-c by x*-\-px-\-q, arranging the product in descending powers of x. ^. Divide 9a*-46*-c* + 46c by 3a-26 + c. 47. Multiply x* — x(o— 1)— I by x* + ax+l. As. Divide o«-166«c* by a-2bc. 49. Arrange the product of x — a, x — b, x — c in descending powers, of X. 50. Divide a*6 + 6*c + c*a — ofe* — 6c* — ca* by a — b, and divide the quotient by a — c. 51. What expression will give a quotient of x*-f-l and a remainder of 2x* — 7x + 6whendividedby 3x»— lOx* — 6 ? 62. Divide x'-y^ + Bt/'- 12y+8 by x — 1/ + 2. CHAPTER XI FACTORING [continued) 89. In Chapter VIII. we have already dealt with the subject of factoring in simple cases. This Chapter will furnish a review of the methods already used, and an extension of those methods to more difficult examples. 90. Type I. Factors common to every Term. When every term of an expression contains the same factor, that factor can be found by inspection (art. 58). Thus, 2xy is a factor of 4:X*y — 6zy* + 2axy, .'. ^x*y — 6xy* + 2axy = 2xy{2x — 3y -\- a), ^ Also, x + y is a factor of aix-\-y)-\-b{x-\-y). When this expression is divided hy x + y, the quotient is a + 6, /. aix+y) + b{x+y) = {x+y){a + b), 91. Type II. Factors by Grouping. When every term has not a common factor, if the number of terms be changed by grouping, we may sometimes obtain a common factor. Ex. 1. — Factor mx-\-nx-\-my-\-ny. (1) mx + nx + my-{-ny^ = x{m + n) + y{m + n), = {m + n)(x + y) (2) mx-\-nx+my-\-ny, = m(x+y) + n(x+y), = (x + 2/)(m + nV Here we changed from four terms to two, and we found a conunon factor in the two terms. The other factor of the expression was then found by division. The two solutions show that different methods of grouping may be Rtaployed. If the first method tried is not successful, try othsrs. 12S / Usually those terms are grouped which contain a simple common factor. In the example we should not expect to be successful by grouping mx with ny, as these terms have not a common factor. Ex. 2.— Factor x^+x^+2x-\-2. Use two different methods of grouping and obtain the factors x+l and a;* + 2. Ex. 3.— Factor {a—h'f—ax-^-hz. {a — b)* — ax + bx = {a — b)* — {ax — bx), = {a-b)»-x{a-b), = {a — b)(a — b — x). Note. — When quantities are enclosed in brackets, the pupil must not forget to verify by mentally removing the brackets. EXBROISB 67 (1-9, Oral) Factor: 1. 3a;-2X 4. b^-5b. 7. a{xi-y)-\-b{x+y). 2. 2a-6. 5. 3a2_i5a6. 8. p{m-n)+(m-n). 3. a2-3a. 6. 6x^y—l2xy^. 9. x{a—b)—2y{a—b). Factor, using two different methods of grouping and verify by multiplication : ^ytO. ax-{-bx+ay-\-by. 11. am—bm-\-an—bn, 12. x'^—ax+bx—ab. 13. bx—ax-j-ab—x^ 14. 2ac+^ad-2bc-3bd. 15. x^+x^+x+1. 16. a=— a2— 3a+3. 17. x—y—xy+l. 18. a;3+4a;2_3x-12. yt^. a^- 7a 2 -40+28. 20. Factor x^+x*—x^—x^+x+l by making three groups each containing two terms, also by making two groups each containing three terms. 21.* Find three factors of 3x^—6x^+3x—6 and of axy—ay—ax-{-a. 22. Find a common factor of am+bm+an+bn and ax-\-ay-\-bx-\-by, and oi x^—x^+x—1 and x^—x^-\-2x—2. I0x^—5zy—6xz+3yz and a^b-\-a^bc—3a^b^—Sab^c. K 130 AIjOEBRA 24, Find a common factor of 2x^—6x2— 3a;+9 and x^— 3x^+2a;— 6, and show that it is a factor of their difference. 25, Factor 48aa;— 56a?/— 356!/+306x. 26, Factor (x+2/)2+4x+4!/ and 2(a-b)^—a+h. 92. Type III. Complete Squares. We have already seen in art. 64 how the square of a binomial may be written down. We have also seen in art. 65 how the square root of a trinomial may be found, when the trinomial is a perfect square. 93. Square of a Trinomial. A trinomial may be squared by expressing it in the form of a binomial or by multiplication. Thus, (o + 6 + c)»=|a + (6 + c)|«, = a» + 2a(6 + c) + (6 + c)*, = o*+2a6+2ac + 6» + 26c + c», .-. (rt + 6 + c)« = a* + 6« + c«+2rt6 + 2ac+26c. Multiply a + 6 + c by a-\-b-\-c in the ordinary way and compare the results. Examine the diagram and see that the same result is obtained. Similarly, (a + 6 — c)«= {a + (fe-c); *, and (a_6-|-c)* = -|a — (6-c)l*. Complete these two in a manner similar to the one worked in full. If we examine these products we see that they consist of two kinds of terms, squares {a"^, b^, c^) and double products {2ab, lac, 26c). We might express the result thus : The, square of any expression is equal to the sum of the squares of each of its terms, together with twice the sum of the products of each pair of terms. In writing down the square, care must be taken to attach the proper sign to each double product. Ex.1. (2x-3?/ + 42)» = 4x»4-9y»+ 162*- 12xt/+ 16x2- 24j^2, 2 a ab ac ab b' be ac be 2 c FACTORING 131 Ex.2. (a-26+c-d)» = oS + 4fe» + c» + d*-4a6+2ac-2a<i-46c + 46d-2cd. Ex. 3. Factor a;* + 42/* + 2*— 4x1/— 2a;24-42/3. This is evidently the square of an expression of the form x+2t/ + 2. Which of these when squared will give the proper arrangement of signs ? Verify by writing down the square. y EXERCISE 68 (1-3S. Oral) What are the squares of : 1. m-{-n. 5. x—2y. 9. x + i 13. a—b—c. 2, m—n. 6, 4x— y. 10. 2a-\. 14. a+b+c+d. 3. 3x+2. 7. 2a-36. 11. X+2/ — 2. 15. p—q—r+8. 4. 3a— 5. 8. Zm—bn. 12. a;-2/+2. 16. x—y-\-z—l. Express as squares : 17. x^+Axy+Ay\ 18. x^- -6XJ/+92/2. 19 . 4x2+4x4-1. 20. 4a2-20a6+2562. 21. 9a' '-12a6+462. 22 . m .'^n'^-^mn+lQ. 23. 4a2+2a+i. 24. 1- -10a+25a2. 25, . a^ '+2o 262+64. What are the square roots of : ^. x^y'^+\0xyz+25z^. 27. 28. 4-20a2-U25o*. 29. 30. m^+n^+p^+2mn+2mp-\-2np. 31. a^+b^+c^-2ab+2ac-2bc. 32. x2_^4j/2+22+4a;y+2xZ + 42/2!. 33. 4a2+62_^9c2_4a6-12ac+66c. Simplify : 34.* (3x-i/)2+(x-32/)2+(2x+32/)2. 35. (a-6)2+(6-c)2+(c-a)2+(a+6+c)2. 36. (x2+x+l)2+(x2-x+l)2. 37. (a— 6+c)2+(6— c+a)2+(c— a+6)2. 38. (3x-22/+z)2-(x-22/+32)2. 16x2— 24x2/ -f9j/2. (a+6)2-2c(a+fe)+c2 K 2 J 132 ^ AL9SBRA Complete the squares by suppijang the missmg terms i 39. x^- . /. , .+25. 40. 4a;2+. . v , . +25y^. 41. a^+iab .- :^\ , . 42>/.'^'*r . . — 12m»+97i*. 43. a2+962+.C * . .-0a0-2ac . V .G'/^C 44. 9x2+.y;. .+. . ... .-6xy-l2xzi.^.C^Z-. 45. Find three factors of 3x'^ + 6a;+3 and of al -4-4026+406*. 46. Factor ^ . c\-^ i^ ->' ~ " ^ j (a+6)2+4c(a+6)+4c^ and \a+b)'—2{a-\-b)(c+d)+{c+d)^. 47. Show that the square of the sum of any two consecutive integers is less than twice the sum of their squares by unity. 48. Divide the sum of the squares of a— 26+c, 6 — 2c+a, c— 2a+6 by the sum of the squares of a—b, b—c, c—a. i i 49. If X + - = 4, find the value of x^ -\ — -. X X* 50. Factor {axYby)'^+{bx—ay)^+c^(x^+y^). 51. Express a^x^+ft^^+a^^+ft^x^ as the sum of two squares 52. Find the value of x2+2/^+22+2x2/+2xz+22/2, when x=a+26— 3c, 2/=6+2c— 3o, z=c+2a— 36 94. Type IV. The Difference of Squares. The product of ohe sum and difference of the same two quantities is equal to the difference of their squares (art. 66) Conversely, the difference of the squares of tveo quantities is equal to the product of their sum and difference (art. 67). Or, in symbols, {a-\-h){a—h)=a^—h^, and a'^-h'^={a+h){a-h). 95. The formula for the product of the sum and difference may sometimes be used to find the product of expressions of more than two terms. Ex. 1.— Multiply 2a— fe+c by 2a—h—c. Here 2a — 6 + c is the sum of 2a — b and c, and 2o— 6— c is the difference of 2a — 6 and c FACTORING 133 rhey might be written {2a — b'j-j-c and (2a — b) — c. The product is therefore the difference of the squares of 2a — 6 and c. /. {2a-b+c){2a-b-c) = (2a-b)*-c*, = 4a*-4a6+6* — c*. Ex. 2. — Find the product of 2x-^y—z and 2x—y-\-z. Here the first expression = 2a;+(?/ — z), and the second =2x—{y—z), .•. the product =(2x)*—{y~z}*, = 4x»-(2/*-22/z+2»), = 4aj* — 2/* + 21/2 — z*. Verify by ordinary multiplication. Ex. 3. — Multiply a—b-\-c—d by a+b—c—d. Note that the terms with the same signs in the two expressions are a and —d These should be grouped to form the first term in each factor. a — b+c — d = {a — d) — {b — c), a-\-b — c—d={a — d)4-(b—c), .. the product = (a— d)* — (6 — c)'. Simplify this result and verify by multiplymg in the ordinary waj: Ex. 4. — Factor p^—4:pq-\-'iq^—z^. Here the first three terms form a square and the expression may be written : [p* — ^q-\-iq*) — x* = {p — 2q)* — x*, = {p-2q+x)(p-2q-x). What two quantities were here added and subtracted to obtam the factors ? Ex. 5.— Factor a^-b^-i-2bc-c^. Here the last three terms should be grouped to form the seconu square. ;. a»-6* + 26c-c« = a*-(6«-26c + c«), = a»-(6-c)», = {a + (6-c)}{a-(6-c)}, = (o + 6 — c)(a — 6 + c). Verify by multiplication. i34 J ALGEBRA Ex. 6.— Factor x^-{-y^—a^-h2-\-2xy+2ab. Evidently three of these terms form one square and the remaining three the other square. The expression = {x* + 2xy + y*)-{a*-2ab + b*), = (x + y)*-(a-b)*, = {x+yi-a-b){x + y-a + b). EXERCISE 69 (1-10. 17-32, Oral) Use the formula to obtain the following products : 1. (2a+3)(2a-3). 3. {xy+5){xy-5). 5. (2m^+3n){2m^—3n). 7. (x+J)(a;-i). 9. {x+y+z){x+y-z). 11.* {a+b—c){a—b+c). 13. {p-2q+3r){p+2q~3r) , 15. {a+b—c+d){a—b-c—d). 16. {a—2b^c—2d)(a-2b-c-^2d). Factor and verify : 17. x2— 9. 20. a%^-x^. 23. l-a262. 26. {x+y)^-'2o. 29. (a+6)2-(c-d)2. 30 32. a^ — 2a6-f-62_c2 35. (4x+3)2-16x2. ^. a^+b^+2ab-c^ 2. (4x-l)(4x+l). 4. {ab—c){ab-{-c). 6. {abc-\-xy){abc—xy). 8. (x2-y2)(x2+t/2). 1«. (a-fe-c)(a-6+c). 12. (2x-|-3i/-5)(2x+3y+5). 14. {\-z+z^)il+x+x'). 18. 4x2-25. 19. a2-46«. 21. 16x''-9?/^ 22. 932-42. 24. 25-x*. 25. (o-6)2-c2. 27. c2_(a+6)2. 28. X2_(y_2)2. 30. (a+26)2-4c 2_ 31. x2 + 2x2/-f-2/2_o2. 33. a2_62_c2_ 26c. 34. a^^b^-^c^^^bc 36. 4— x24-2x2/- -y'- 37. a^-x^+2ay+y^. '2cd. 39. a2— 62-(-c2— d2— 2oc— 26d, 40. o2-2rf-hl-62-^26c-c2. 41. x«-x2-4— 2x22/2_4x-f-yS 42. 4x2_4x_2/2+4ay— 4a2-(-1.^3, 1 +a2— 62— 4c2+46c— 2a. 44. Find three factors of 2x2—8, a^ — a, a*— x*. FACTORING 136 45. Find three factors of ba^—\0ah+5b^-20c^ and of (a;-36)3— 462x4-1263. 46. Find four factors of a^b^—aH^—bH'^-\-c'^d'^ and of (a2+62-c2)2-4a262, 47. Factor a'^x'^ —b'^y'^ ^2aA^x-\-c'^ and w^— Gm^n^+w^— 2m». 48. Find the simplest factors of 3x^—2^^—30; +2 and of a;*— a;3_9x2+9x. 49. Simplify (a— 6+ c)(a— 6— c) + (a+6-c)(a— 6+c). 50. Arrange x^{x^—a^)—y\y'^—a'^)-{-2xy{x^—y'^) so as to show that x^—y"^ is a factor of it, and thus find the simplest factors. 51. Use. factoring to simplify : ' '^Af '■''^^''^^'-(1) (a2-3a+l)2-(a2-3a)2. ' - (2) (a;-22/+32)2-(32-x+22/)''=. (3) (a2-3a-4)2-(a2-4)2. (4) (5x2-2a;2/+?/2)2_(5x2+2a;y-2/2)2. 52. Multiply aArb-\-c by a + 6— c and a— 6+c by a—b—c and use the results to obtain the product of (a+b-\-c){a-\-b—c){a—b-\-c){a—b—c). 53. Show that x{y^--z^)-\-y{z^—x^)-\-z(x^—y'^) is equal to {x—y){y^—z^)-(x^—y^){y—z) and then find the factors of this expression. 54. Arrange a{b^—c^)-^b(c^—a^)+c{a^—b^) in the form o(62-c2)-6c(6-c)-a2(6-c) and thus obtain the factor b—c. Find the other two factors. 96. Type V. Incomplete Squares. We have already factored many expressions which were seen to be the difference of two squares. Sometimes the two squares of which an expression is the difference are not so easily seen. 136 ALSEBRA Ex. 1.— Factor .r^+.r^/z^+y* This expression would be the squar*^ oi a-' + y* if the middle term were 2x^y*. We will therefore add x^y* to complete the square and also subtract x^y* to preserve the value of the expression. Then x* + x^y* + y* = x*+2xh^* + y*-x*y*, = (x* + y*)*-{xy)', = {x* + y* + xy){x^ + y*-xy), .: x*+x*y* + y* = {x* + xy + y^){x*~xy + y*). In order that this method may be successful, it will be seen that t1ie quantity we add to complete the square must itself be a square. Thus, to change a* + ab + b* into a*-(-2a6 + 6* — a6 is of no value as 06 is not an algebraic square. Ex. 2.— Factor a*+4t*- This can be made the square of a' + 26* by adding 4a*6*. ,\ a« + 46« = a« + 4a*6«-|-46*-4a«6». Ck>mplete the factoring and verify by multiplication. Ex. 3.— Factor 4m<— IBw^n^+gn*. What must be added to make it the square of 2m* — 3n* ? Complete the factoring. Try to factor it by making it the square of 2m* + 3n*. Ex. 4.— Factor a*-\rb*-\-c*-2a%^-2b^c^-2c^a^. How does this expression differ from the square of a*-f-6* — c' T Express it in the form (a* + 6* — c*)* — 4a*6*, Write down the two factors and see if you can factor each of them again and thus obtain the result (a + b + c)(a+b — c){a—b+c)ia—b—c). ^ Factor and verify : EXERCISE 70 1. a*+a2+l. 2. x*+x24-25. 3. x*+7x«+16. 4. a:« + 2a;V+92/*- 5. 4a*+L 6. 9x«+8xV+16?/*. 7. 46*-136*-f-l 8 9a*-15a24-l 9, 9a*— 5202^2 4-646* FACTORING 137 10. 26a;*— 89z^*+642/'= li- x*+y*—\\x^y^. i^^x^—lx*+l. 13.* Find three factors of 2a;*+8 and x^+z^+x. 14. Find four factors of 9a*—l0a^b^+b^, 15. Find three factors of x^+x^+l- 16. Find four factors of a*+b*+c*-2a^b^—2b^c^—2c^a^ by completing the square of a^—b^-\-c^r 17. Factor (a+l)*+(a2— l)2+(a— 1)*. 97. Type VI. Trinomials. We have already dealt with the factoring of expressions of the type x^-{-pz-\-q, where the coefficient of the first term is unity (art. 61). We now wish to factor expressions of the t;^^e 'mz^-\-px-\-q, where m is not necessarily unity 98. First Method, by Cross Multiplication. Ex. 1.— Factor 2x^-\-lxy-{-3y\ The product of the first terms of the factors is 2a;*, and therefore the first terms must be 2x and x ; similarly, the last terms must be 3y and y and the signs are evidently all positive. .'. the factors must be 2x+3y 2x+ y z+ y or x+Sy It is seen, by cross muitiplication, that the coefficient of xy in the first product is 3 + 2=5, and in the second is l-i-6 = 7. .'. the correct factors are (2x-\-y){x-ir3y). Ex. 2.— Factor 3x2— 7a;— 6. Here the numerical coefficients of the first terms of the factors must be 3 and 1, and of the last terms may be 6 and 1 or 3 and 2. Since the third term is negative, the signs of the second terms of the factors must be different. The possible sets of factors, omitting the signs, are : 3* a 3a; 2 3a; 6 3a; 1 *2 X 3 X I x 6 138 ALOE BRA Since the signs are different tor the last terms, when we cross multiply to find the ooefficien*^ of x in the product, the partial products must be subtracted It is easily seen that the second arrangement is the only one from which 7z can be obtained. Since the middle term is negative, the larger of the cross products must be negative. .-, the factors are (3x+2)(x-3). This method is liable to be found tedious when the coefficients have a number of pairs of factors, but in ordinary cases the pupil will find little difficulty after he has had some practice in the work. 99. Second Method, by Decomposition. In the process of multiplying two binomials hke 2x+3 and 3x-\-5, we have (2a;+3)(3x+5)=3:c(2a;+3)+5(2a;+3), =ex^-\-9x-\-l0x-\-15, = 6x2+19x+15. If we wish to factor a trinomial like Gx'^+lQ^+lS, we may do so by reversing the process. Thus, 6x2+19a:+15=6a:2+9a;+10x+15, = 3x(2.c+3) + 5(2a:+3), = (2x+3)(3x+5). The only difficulty in this method is in finding the two terms into which the middle term, 19a:, should be decom- posed. This difficulty may be overcome in the following way : {ax-\-b){cx-{-d)—acx^-\-x{ad-\-bc)-\-bd. Note that the product of the two terms in the coefficient of z, ad and be, is the same as the product of the coefficient of x^, ac and the absolute term, bd. In the trinomial 6z2-|-i9x-|-15 above, the product of 6 and 15 is 90 and the two factors of 90 whose sum is 19 are 9 and 10, which shows that the middle term, 19a;, should be decomposed into 9x4- 10a;. FACTORING 139 / Ex. 1.— Factor 6.c2+13a:+6. The produc*^ of the coefficient of x* and the absolute term is 3& The two factors of 36 whose sum is 13 are 4 and 9. .-. 6x*+13x+6 = 6x» + 4a; + 9x+6, = 2x(3x+2) + 3(3x + 2), = (3x+2)(2a;+3). Ex. 2.— Factor \2x^-\lx-5. Here we require two factors of —60 whose sum is —17, and thej are evidently —20 and 3. .-. 12x*-17x-5=12a;«-20x+3x-5, = 4a:(3x-5) + (3x-5), = (3a;-5)(4x+l). BXBRCISn 71 (1-18. Oral) Factor and verify : 1. a:2^4x+3. 4. a2-lla+18. 7. x2_i5a.^l4_ 10. l-21x+38x2, 13. x2--4x-5. 16. 2/2_4y_2i. 19. 2x2+5x+3. ^^^0^2. 8x2+x-g. 25. 4x^+x— 5. 28. 1062-896 .^. 2. a2+lla4-30. 5. x2-14a:+48. 8. a'^b'^-^ab+Q. 11. x2-6x2/+8i/2. 14. a2-9a— 22. 17. l-2a-loa2. 20. 4x2 +8x2/ +32/2. 23. 3x2-35-2, 26. 1562-196-8. 9. 29. 9x2— 31xi/+12i/2 3. 2/2+8i/+15. 6. l+5x+6x2. 9. a2-15a+56. 12. a2-13a6 + 3662. 15. x2-28x-29. 18. a2— aw- 2i/2. 21. 9a2-18a6 + 862. ,24. 6a2_a-2. 27. 10x2-23x-5. 30. 10a2-29a6 + 1062. 33. X— 5x2+6a;^ 36. 9a*-10a2+l. Find the simplest factors of : fc^l.* 3x2-3x-216. 32. 2a2+8a+6. 34. X*— 5x2+4. 35. a^-Kki^+^a. ^y^. (x2+4x)2-2(x2+4x)-15. ^. (x2-9x)2+4(x2-9x) - 140. 3**. Without multiplying show that (x2-x-2)(x2+2x-15) = (x2 + 6x + 5)(x2-5x+6). 40. An expression is divisible by x— 2, the quotient being x2— x— 6. Show that it is divisible by x + 2 and find the quotient 140 ALGEBRA 41. Show that the product of 6x2- 13a; -(-6 and 2x^-lx + 5 is divisible by 3x^— 5a; + 2 and tind the quotient. 42. If 3x^-\-ax—\'l is the product of two binomials with integral coefficients, find all the different values that a may have. 43. By factoring, find the quotient when the product of 6a^ + lab-20b^ and 22a^-l^ab-Ub^ is divided by 40^— 4a6 — 356^ 44. Factor x'^-{-5xy-\-^y^-\-x-\-y. 45. Factor Za^—ab—2b--\-%a+4:b. 46. Divide the product of x^-\-Zx+2 and a;*— 1 by the product of x2+2a;+l and x^+x — 2. 100. Type VII. Sum and Difference of Cubes. Divide x^-^y^ by x-\-y, x4-y)x^ +y^{x*-xy-\-y* and x^—y^ by x—y. ^ +^ ^ .-. a53+i/3=(a5+i/)(a52-aci/+i/2), -^v _ ^ and x^—y^={x—y){x^+xy-\-y^). ^v ^ Examine carefully the signs in these +^ +y factors. "^'^y It is thus seen, that the sum of the cubes of two quantities is divisible by their sum, and the difference of the cubes iif divisible by their difference. The quotient in each case consists of the square, product and square of the terms of the divisor., with the proper algebraic signs Ex. 1.— Factor 8a^+21b^. Here 8a* = (2a)» and 276» = (36)», .'. the expression may be written (2a)' + (36)', ' .'. the first factor is 2a + 36 and the second is (2a)»-(2a)(36) + (36)* or 4a*-6a6 + 96>. .-. 8a3 + 2768 = (2a + 36)(4a»-6a6 + 96»). Ex. 2. — Factor a^x^—64y'^. o«a;»-64y« = (az)»-(42/*)», = (a!e-4v*Xo*a;» + 4<ftB|f s+ !8y«> FACTORING 141 Ex. 3.— Factor .r«-?/8. This may be expressed as the difference of two squares or of two cubes. .-. x8-2/« = (a;»)»-(?/3)«, or (a;*)»-(2/»)3, = {x^ + y^){x^-y^), or {x*-y^){x^ + x^y'^^y'^). Complete the factoring by each method and decide wliich you will use, if you have the choice, as here. EXERCISE 72 (1-12, Oral) State one factor of : 1. a^+b\ 2. x3+8. 3. a;= '-27. 4. 1000-a». 5. a;3— 642/3. 6. 27-63. 7. 8a3+I25. 8. 125a3-863. 9. l-27a;3. 10. 343x3-8. 11. (a+6)3+c3. 12. (a-fe)3_c3. Factor and verify 13-21 : (aH, a^+21. ^,14. x3-8?/3. ^15. 8a3+l. 10. 27x3-64!/3. ,17. 8-27a3. .18. 1000x3-2/3. ^9. a3+6«. 20. x«-63. v^- a3_7/6. 22.* 2a3-16. 23. 81+3</3. 24. a*+a. 25. a%-\-h*. 26. a8+6«. 27. (a;+2/)3+a3. 28. (a;-2)3+8. 29. (a-6)3+a3. ® (a-6)3+(a+6)3. 31. What is one factor of {2x—yf—{x-2yf ? 32. Show that (2a-36)3-(-(3a-26)3 is divisible by a—h. 33. Factor (a2-26c)3+863c3 and 21xhjh-yh*. 34. Find six factors of a^^—b^\ 35. Find two binomial factors of (2x2-3x+3)3— (x2-22;4- 5)3. 36. If X + 1 =2, find the value of x3 + i . X X3 37. By factoring show that {a+b)*—3ab{a + b)'^={a+b){a^+b^). 101. Type VIII. The Factor Theorem. What are the values of 0x5. axO, 0x(-4), ^xO, -1000x0^ Itt ALGEbBA If one of the factors of a product be zero, the product must also be zero. If the product of two numbers be zero, what can we infer 1 If a6=0, it follows that either a=0 or 6=0. If (x— 3)(x— 4)=0, then either x— 3=0 or a:-4=0. Since (x-2)(x2-7x+12)=2;3— 9a;2+26j;-24, .'. x^— 9x2+26x— 24 must be equal to zero when x=2, for then one of its factors, x—2, is zero. If we substitute 2 for x, we see that this is true. a;3-9x» + 26x-24=2»-9 . 2» + 26 . 2-24, = 8-36 + 52 -24 = 0. Conversely, when any expression becomes zero when x=a, then x—a is a factor of it. Substitute x=3 in x^ — &x*-^\\x — Q and it becomes 3»-6 . 3»+ll . 3-6 = 27-54 + 33-6 = 0, a; — 3 is a factor of x' — 6a;* + llx — 6. Divide it by x — 3 and the other factor is x* — 3x + 2. .-. x»-6x*+llx-6 = (x-3)(x*-3x+2), = (x-3)(x-2){x-l). If x-|- 1 is a factor of an expression, the expression must be equal to zero when x= — 1, for then x+ 1 = 0. Thus, x+1 is a factor of x' — x*— lOx— 8, since (-!)»-(- l)»-t0(-l)-8=- 1-1 + 10-8 = 0. Divide by x+ 1 and complete the factoring. Any expression is divisible by x—a, if it vanishes {becomes zero) when a is substituted for x. This is called the factor theorem. Show that x — a is a factor of x' — 7ax*+ 10a*x — 4a'. Show that x + a is a factor of 5x' + 6x*o+ llxa*+10a*. Ex.— Factor x^— 9x+10. If it has a binomial factor it must be of the form x±l, x±2, x±5 or x+10. Testing for these factors we find that x— 2 is a factor, .-. x«-9x+10 = (x-2)(x* + 2x-5). The factoring is complete as x* + 2x— 5 has no simple factora^ FACTORING 148 102. Special Case. It is easy to see when x— 1 is a factor of any expression, for when 1 is substituted for x, the value of the expression becomes equal to the sum of its coefficients. Thus, ifx=l x»-2x»-19x + 20, = 1 -2 -19 +20 = 0, .'. X— 1 is a factor. Complete the factoring. Similarly, x — a is a factor of 3x'— 16a;*a — 7a;a* + 20o', since 3-16-7 + 20 = 0, and a-b is a factor of a«-6o*6 + 3o6* + 26», sine? 1-6 + 3 + 2 = 0. EXERCISE 73 Each of these expressions is divisible by x — 1, x—2 or x — 3. Find all the factors of each and verify. 1. z'-I0a;2+29a;-20. 2. x^-Zx'^-\2x+\i:, 3. x3+5a;2— 2x— 24. 4, x-—^x^-\-x+^. 5. 2a;3-7x2+7x-2. 6. 4x^-9x2- lOx+3. Factor : ^ 2x3-llx2+5x+4. ,8. x'-2x2-x+2. (^ x3-7x+6. ^^. x3-19x+30. 11. o3+a2-10a+8. ^f2^ a'^-'iab^-2b^. / 13. Show that x+2 is a factor of x'— x*— x+10. 14. Show that x+o is a factor of x'+7x%+9xa^ + 3a^ 16. Show that x+3, x+4 and x— 7 are the factors of x'— 37x— 84. 16. If x^— lOx+a is divisible by x+2, find a. 17. Show that a—b is a factor of a^+'ia'^b+ab'^-%^, and find all the factors. 18. Noting that x^- 2x— 3=(x+l)(x— 3), show that x^— 2x— 3 is a factor of x*— 4x3+2x2+4x— 3. 19. Show that a—b, b—c and c—a are factors of a(62_c2)+6(c2-a2)+c(a2-62). 20. If X— 1 and x—2 are factors of x^— Sx^+ax+ft, find a and b. 21. If px3-3x2+gx-10 and qx^+2x^-\lx+p are both divisible by x—2, find p and q. 144 ALOBSBA 103. Equations Solved by Factoring. We have seen that if then z— 3 = or x— 4=0. Thus the equation (x— 3)(x— 4)=0 is equivalent to the two simple equations a:— 3=0 and x—4=0. But if 2—3=0, x=3, and if x — 4=0, a;=4, .'. the roots of the equation (x— 3)(x— 4) = are 3, 4. The truth of this may be .seen by substitution. If a;=3, (x-3)(x-4) = (3-3)(3-4) = 0x -1=0. Ifa; = 4, (x-3)(x-4) = (4-3)(4-4)=lx = 0. Since (a;-3)(x-4) = x*-7x+ 12, the given equation may be written x»-7x+12 = 0. 104. Quadratic Equation. Any equation which contains the square of the unknown and no higher power is called a quadratic equation or an equation of the second degree. The preceding shows that if we wish to solve a quadratic equation we may do so by finding, by factoring, the simple equations of which it is composed. The simple equations when solved will give the roots of the given quadratic equation. Ex. 1.— Solve x2-6a:-7=0. Factoring. (x — 7)(x+l) = 0, .-. X — 7 = or x+i=0, x=7 or — 1. Verification : if x= 7, x*-6x-7 = 49-42-7 = 0, if x=-l, x*-6x-7= 1+ 6-7 = 0. Ex. 2.— Solve 3x^-\-lx=6. Transpose the 6 so aa to make the right-hand side zero, as in the previous problem 3x»-f7x-6 = 0, (3x-2)(x-l-3) = 0, .-. 3x-2 = or x + 3 = 0, x=f or ^3. Verify both of these roots. FACTORING 145 Ex. 3. — Form the equation whose roots are 2 and —5. The required equation is at once seen to be a combination of the two simple equations a;--2 = and x-t-5 = 9, and therefore is {x—2){x-\ 5) = 0, or x» + 3a;— 10 = 0. Ex. 4. — If 2 is a root of the equation x3 + 3a;2-16x+12 = 0, find the other roots. Since 2 is a root, then x —1 is a factor of x^-\-^x'^ — \Gx-\-\'i and the other factor, found by division, is x^-f-Sx— 6. .-. a;» + 3a;»-16x+12 = (a;-2)(a;-l)(x+6) = 0, X — 2 = or X— 1 = or x+6 = 0, x=2 or 1 or —6. .'. the other roots are 1 and — 6. BXBIROISEI 74 (1-16, Oral) To what equations of the first degree is each of the following aquivalent : i. (x-l)(x-2)=0. 2. (x-3)(x+5)=0. 3. x(a;-5)=0. 4. (x-l){x-2)(x-3)=0. 6. x2— 4=0. 6. x2— 4x-}-3=0. 7, x2_|_5a;_|_6=0. 8. a;2_a:-20=0. 9. x2+3ax+2o2=:0. 10. x2-6x- 1262=0. State the equations whose roots are : 11. 2 and 3. 12. 4 and —5. 13. —2 and —4. 14. a and h. 15. 2, 3 and 1 16. 4, 5 and -6. Solve and verify : 17. x2--8x+15=0. 18. x2+8x4-15=0. ■i». x2+2x-15=0. ^. 3x2-8x4-4=0. x2-2x-15=0. 4x2_2x-2=0. 23. 2x2-|-x=15. 24. x(3x-l)=10. ^6. x»-«=0. 26. x'— ox+fcz— a6=0. 146 ALQEBitA 27. If 2 is 3 root of j-'- 19j-+30 =0, find the other roots. 28. Solve z3— 6x2+llx— 6=0 and ix^— 12x=^4-llx-3=0 (Note that the sum of the coefficients is zero.) 29. The sum of a number and its square is 42. Find the number. 30. The sum of the squares of two consecutive numbers is 61. P'ind them. 31. The sides of a right-angled triangle are x, x-\- 1 and x+2. Find x. 105. Notes concerning Factoring. The subject of factoring is one of the important parts of algebra, as it enters into so many other processes. We have already had examples of its use in solving equations and in performing operations on fractions. In the preceding exercises, in this Chapter, the expressions to be factored have been classified for the pupil. In the practical use of factoring, however, he must determine for himself the particular method to be used. This is usually done by determining the type or form to which the expression belongs. The examples in the review exercise which follows will give the required practice. The types which have been discussed in this Chapter are here collected for reference : I. ax-\-a(j. (Common factor in every term.) II. ax-\-ay-\-hx-\-hy. (Factored by grouping.) III. x^±2xy-\-y'^. (Complete squares.) IV. a^—h^. (Difference of two squares.) V. x*'-\-x^y^+y'^. (Incomplete squares.) VI.' ax'^-\-bx-\-c. (Trinomials.) VII. X^ ± y^. (Sum or difference of cubes.) VIII. Factored by the factor theorem. FACTORING 147 BXERCISE: 76 (Review of Chapter XI) 1. State the squares of a-\-b, a — b, x—3y, 2x—\, 3x — 5. 5a-\ 2b !ix—4:y,7a—Z,a*—l,a-\ 2. State the squares of o + 6 + c, x-\-y — z, a—b — c. 3. Writedown the products of x(a — b), a(o — 6 + c), {x-\-l)(x-\- ^ [x-3)(x-5), (2a;-3)(2x+3), (ax-6)(ax+7). Use short methods to find, in the simplest form, the value of : 4.* {x + a + b){x + a-b) + {x — a-b){x-a + b). 5. (x* + x+l)(x* + x-l)-(x*-a;+l)(x*-x-l). 6. (a + b + c)* + {a + b-cy + {a-b + c)^ + {a-b-c)K 7. (2a+36-c)« + (3a + 6-2c)« + (a-26-|-3c)*. 8. 9999«-9998*. 9. 57432-4257*. 10. 503x497-502x498. 11. (a + 99)«-(a+98)2. _^. Find the simplest factors and verify 12-29 : ^^■ 1^12. x*-x-42. ^^\. x' — 4x. y-^Q. X* — a + ax — X. i.,^8. x» + 5x«-4x-20. ^^20. 1 5a* + 32a + 9. 22. x* — 4tx*. 24. (x-3)* + (x-3)(x + 4). 26. l + 2a6-a»-6*. 28. abc* + a*cd + abd^ + b^cd. >80. 4(x-2)*-x+2. 32. (a+26-3c)*-(3a + 26-c)« 34. 108a«-500. 36. x«-7x*-18. 38. X--X2/-1322/*. 40. x^ + y^ + 3xy{x + y). 42. o*-46«-3a-66. <i4. a* + 2ab + b*-\-ac + bc. 45. o*-2o6 + 6*-a+6. L 2 ^3. x»-3x*-x+3. >5. a* + a-56. , 27x*-12i/». /^'^ ^9. x»-3x« + 2x. • 21. 343 -x«. ,23. 18x»-l-48x+32. 25. 15x*-152/«-16xj/. 27. x(x-2)+2/(x-2)-x + 2. 29. 25x*2/ — 40x^2/2+ I6x*t/*. 31. 24o«-3a6». 33. 12x*-x-20. 36. x^ + x-^-y^ + y. 37. x'-y'-2x^y + 2xyK 39. (a + c)(a-c)-6(2a-6). 41. ax*-x(3o6 + 2) + 66. 43. 2x{2x + a) -y{y + a). 148 ALGEBRA 46. x*-\-x* + x-y* — y*-y. 47. a*b - a*b' - a»b* + ab*. 48. ia*-25b* + 2a + 5b. 49. 8(a + 6)»-(2a-6)». ,60. x* + y*-18x*y*. 61. a*-a«-9-2a«6« + 6*+6a. ^62. x»-Ux*+7x+3. 63. 3a»-5a*-8a+10. 64. x«c»-c» + x*-l. 65. a'-a'' + 8a«-8. 66. Show that a — b + c is a factor of (2a-36 + 4c)» + (2o-6)». 67. Factor 4a« — 37a*6* + 96«, (1) by cross multiplication, (2) by completing the square of 2o* — 36*, (3) by completing the square of 2a» + 36». 68. Without multiplication show that (a;»-4a;+3)(z«-12x + 35) = (x*-6x + 5)(x«-10x + 21). 69. Make a diagram to show the square of a + ft-f-c + d. 60. Factor (a-b)(b*-c*)-{b-c)(a*-b*). 61. Find the factors of 6x' — 7x*— 16x-f- 12, being given that it vanishes when x = 2. 62. Find four factors of (x*-5x)»+ 10(x* — 5x)-|-24 and of (x*-6)*-4x(x*-6)-5x«. 63. Use the factor theorem to solve x*-31x + 30 = and x«-43x» + 42x=0. 64. If two numbers differ by 6, show that the difference of their squares is equal to six times their sum. 65. Find the quotient when the product of x* — {b — c)x — bc and x* — (c — a)x — ca is divided by x' + (a — 6)x — a6. 66. Multiply a*-b*-c» + 2b€ by °+^ + ° . a + b — c 67. If x* + x^ -\- ax* -\- bx — 3 is divisible by x— 1 and x+3, find a and 6 and the remaining factor. 68. Factor 2x* — ax + bx — ab — a*. 69. Express a*b*-^-c*d* — a*c* — b*d* as the difference of two squares in two different ways. 70. Factor a*-{-b* + c*—2a*b* — 2b*c* — 2c*a* by completing the square of a* — b* — c*. 71. Find four factors of {a*-b* — c* + d*)* — 4{ad-bc)*. CHAPTER XII SIMULTANEOUS EQUATIONS {continued) 106. In Chapter VII. the solution of simple examples of equations in two unknowns has been considered. The method there followed was to make the coefficients of one of the unknowns numerically equal by multiplication, and then that unknown was eliminated by addition or subtraction. Other methods of eliminating one of the unknowns are useful in certain cases. 107. Elimination by Substitution. Ex.— Solve x—2y= 2, (1) 5z-^ly=18. (2) Froin(l), x = 2 + 2y. (3) Substituting this value of x in (2), 5(2 + 22/) + 72/ = 78, .-. 10+102/ + 72/ = 78, 172/ = 68, 2/= 4. Substituting 2/ = 4in(3), a;=10. Here we eliminated x by finding the value of z in terms of y from (1) and substituting that value in (2). We thus obtained an equation which contained only the unknown y. This is called the method of elimination by substitution. We might take the value of x from (2) and substitute in (1). Thus from (2), 5x = 18-ly, .'. x= ^^T^^ ' Substituting in (1), ^^~S^ — 2y = 2. Complete the solution and verify the roots, 140 150 ALOE BRA The value of y might have been found from either equation and substituted in the other. _, o Thus from (1), 2y=-x-2, :. j/=— — . Substituting in (2), 5x + "^^^s^^^ = 78. Complete the solution. Solve also by finding y from (2) and substituting in (1). If the four solutions be compared it will be seen that, in this problem, the first is the simplest. In solving equations with two unknowns, the pupil should examine them carefully and choose the unknown which he thinks will be the simpler to deal with. 108. Elimination by Comparison. E^-Solve 2a;- 3?/= 7, (1) 3x-\-5y=39. (2) 7 + 3y _ 39-52/ 2 3 ' .-. 3(7 + 32/) = 2(39-51/). Complete the solution and verify theNi;Dots. Here we effected the ehmination of xM^ comparing the values of x from the two equations. This is called the method of elimination by coln^arison. We might have compared the values of y obtained from the two equations. Solve it that way. 109. Three Methods of Elimination. We have illustrated three methods of elimination, by addition or subtraction, by substitution and by comparison. When no particular method is specified, the pupil is advised to use the first method as no fractions appear in it. SIMULTANEOUS EQUATIONS 16i EXERCISE 78 (1-6, Oral) State the value of eaoh unknown in terms of the other in : 1. x+y^5. 2. X— y=3. 4, Zx—y^%. 5. 2a;+3;/=12 Solve by substitution and verify : 7 \y x+2y=\%, 2x+oy=i\. 10. 2z— 32/=14, x-5y= 0. /S. Zx+ y^ 7, 4x+3y=ll. 11. 3a;— 4i/=10, u^ 2x+&y=\\. Solve by comparison and verify : 14. 13. x+3y= x-\-by-- 10, 14. 2x-rJ/= 3x— !/= = 26, = 14. Solve by any method and verify 16. \x-\y=2. 19. 2 3' !=■'■ ,17. 20. Zx^2y, hx=ly-2. 5 ^3 3. 6. 12. 15. 18. x+2y-ll. 5a;— 4?/= 19. 2z— t/=19, 5a;-32/=46. 8a;+ 9?/= 7, 10x+21?/=12. 3a;+4!/=10, 4a;-3i/= 5. 2/= fa;- ■-\x-\-&. a;+ sy. 11|. 2^ 3y—~x=x, 3x-l = 1. 22. 3a; + 2 23 y+7 .11 2y + ?±i^ = 10. x-52/+3=2a;-82/+3=7a;- lOy+16. 24. (x-l)(2/-2)-(y-3)(x+l) = 17, (x-3)(2/-5)-(x-5)(2/-3) = -22. lx+-21y+-52=01a;+01?/+3-0. 25. 26. 27. x+5 = 3(?/-3), 5x- 11 4 , 2x 4x- -3y— 5 _ 4 lx-2y- 23 9 2x- + 19. 9y 28. If the sum of two numbers is ^ of the greater number, the diSerence of the numbers is how many times the less ? 162 ALGEBRA 110. Eqiy^iops with three Un]^wns. Ex.— Solve 2x-\-3y—4z=12, • (1) 3x- y+22=15, * (2) 4a;+ y-3z=19. '^ t (3) This system of equations differs from the preceding by containing three unknown quantities. If we can obtain from these three equations, two equations containing the same two unknowns, the solution can be efiFected by preceding methods. How can we obtain from (1) and (2) an equation containing X and z only ? How can we obtain another equation from (2) and (3) containing x and z only ? Perform these two eliminations and find x and z from the resulting equations. Now find y by substituting in any one of the given equations and verify by showing that the values you have found for x, y and z will satisfy all of the given equations. The solution might be written in the following form : Eliminate y from (1) and (2), .-. llx+22 = 57. (4) (2) „ (3). .-. 7x- 2 = 34. (6) 2 » (4) „ (5). a;= 6. Substitute x=5 in (4), 2= 1. „ x=5 and 2=1 in (1), 1/= 2. .•. x = b, 2/ = 2, 2= 1. Of course it will be seen that any other unknown might have been eliminated twice from two pairs of the equations. Thus we might have eliminated z from (1) and (2) and also from (1) and (3), and thus obtained two equations in x and y. We might then have completed the solution as before. Solve the equations by this plan. Also solve them by two eliminations of x. Which letter do you think is easiest to eliminate twice T Note that the solution is completed only when the values of all of the unknowns have been found. SIMULTANEOUS EQUATIONS 163 EXERCISE 77 (1-4, Oral) 1. What operation will eliminate both x and a;+?/-|-z=35 (1) y from (1) and (2)? What is the value of 2? x-\-y—z=25 (2) of x+yl x-y-\-z=\5 (3) 2. What operation will eliminate both y and z from (2) and (3) ? What is the value of a; ? of y—z ? 3. How can you eliminate both x and z from (1) and (3) ? What is the value oiy t of x-\-z ? 4. In No. 5 below, which letter is simplest to eliminate from two pairs of the equations ? Which in No. 6 ? Which in No. 7 ? Solve and verify : 6. x+2y+ 3z=16, 6. 2a:— i/+3z=- 7, ^\ x+Zy+ 4z=24, 3a;+s/— 42= 7> a; -t-42/-|- 102=41. 6x— !/+52=21. 7. 4x-3i/+ 2=10, 8v x+y— 2=16, ^ 6x— 5?/+22=17, -' x—y-{- z= 4, a;+ i/+ 2= 8. a;+y+22=22. 9. x+22/+3z=32, 10. a;+?/=25, 4a;— 52/-l-6z=27, 2/+z=75, .^ 7x+8y-92=14. 2+a;=70. a;+22/=12, ^. 3(2-1) =^2{y-l), 32/+42= 2, 4(2/+x) =92 —4, 5z-2x=-21. 7(5x-32)=2y -9. J 13. 3+x=5+4!/, ^. Ja;-f i2/+ Jz =36, ' 2+x=3i/, ^ \ lx+^\y+.}QZ =10, 7?/=z+2. ira;+ i2/+Az =43. •^^^^ 2 + 3+4-3 + 4 + 5-4 + 5 + 6-^- 'r 16.* If x-\-2y=25, ?/+32=55, z+4a:=35, find the value of x+y-\-z. 17. If x—y-\-z=9, 2x-\-y=8, 1/— 42=5 and x+y-\-z-\-w=l2, find i«. 18. If ax^—hx-\-c is 8 when x=l, 8 when a;=2, and 10 when a;=3, find a, 6 and c. V.f.if 154 ALGEBRA 19. If ax^+bx+c is 9 when x=l, —3 when x= — 1, 18 when z=2, find its value when x = 3. 20. Determine three numbers whose sum is 9, such that the sum of the first, twice the second and three times the third is 22, and the sum of the first, four times the second and nine times the third is 58. 21. If + 6=12, 6+c=15, c+d=l9, find a+d. 111. Special Forms of Equations. Two equations of the first degree in x and y will usually determine the values of X and y. Consider the following sets of equations : (1) 2x-3^=10, (2) 2x-3y=l0, (3) 2x-3y=l0, 4j-+5j/=42. 4x-6.y==20. 4x-6!/ = 30. In (I), if the two equations are solved in the usual way we find that x=8, y=2 will satisfy both of the equations, and no other values of x and y will satisfy them. We therefore say that these equations are determinate, that is, they determine the values of x and y. In this ca.se the second equation can not be deduced from the first, nor the first from the second. We therefore say that the equations are independent. In (2), the second equation may be deduced from the first by multiplying by 2. These equations are dependent and not independent as in (1). Any number of values of x and y will satisfy both equation^ because any values which will satisfy the first will also satisfy the second. These equations are therefore indeterminate. In (3), if the first equation is true, the second can not be true. They are therefore said to be inconsistent or impossible, and no values of x and y can be found to satisf}' both of them. We thus see that tioo equations in two unknovms can have a definite solution only when the equMions are independent and consistent. In this set of equations, the third may be Zx-\-2y— z= 5, obtained by adding the other two. They are 4x— y-\-Zz = 2Q, therefore dependent equations and consequently lx-\- y + 2z = 25. indeterminate . SIMULTANEOUS EQUATIONS 156 EXERCISE 78 1. Find three pairs of values of x and y which satisfy the equation 2x-Zy=\2. 2. Solve 2x+3?/=13, bx—y=24^. Is it possible that 2x+3!/=13, 5x—y=24: and 4x+5?/=19 can be true at the same time ? 3. What is peculiar about the equations i^x-\-y=\l, %x-\-2y--^5 7 Also about 8a; +12^ =60, 6x+9i/=45 ? 4. Show that the equations x+z+4=32/, 2x+z=2y+%, 2x+t/=10, are indeterminate. If 2=5, solve the equations 5. Find two solutions of the simultaneous equations : x+z/+2=10, ^x—2y—z=l. For what values of a will the following sets of equations be consistent : 6.* 3x— y= 5, 7. 3x+2«/= 7, 8. 9a;— ay=Q, x+2y=25, I0x-'iy= 2, 3x- y=2, x+4y= a. 3x+ay=ll. ox—^y=^. 9. Show that these equations are inconsistent : 2x+3y-3z=20, 3x+7y—2z=5, x+2y—z=6. 112. Special Fractional Equations. Ex.— Solve --- = 11, (1) z y 4 2 - + - = 21. (2) X y Here we could obtain the solution in the usual way by removing the fractional forms, by multiplying each equation by xy. See if you can complete the solution by this method. It is simpler, however, to eliminate v from the equations as they dtand. 166 ALOE BRA Thus, multiplying (2) by 4 and adding 95x = 19, '^ = 95, 1 8 Substitute x = - in (1) and 15 5 y 1 11. 2/ = 2. The solution, therefore, is x = l, y- Verify this result. Solve and verify : ^ y 24_21_j X 1/ 2, yi _4 y ii+i * y 101. EXERCISE 79 ^ ^ y 5 6 y 29, 2. + 2t/ = 15, --3y = 6. 1^ X y 7_9 X 2/ 1 - - = 3. ! + ?=!.. X 1/ ?-? = 2. y 2 ^-1 = 17. 9. 10. 11. 3y—5x=xy, 2y+3x=2exy. 5^3^13575^3^ X J/ X J/ 3x + ? y x^22/ = i^ + l = 2x^3y 12x + - + 14 = - - 2x - 14. y y 4.^1 + ' X u 122 X ?/ 82 + 17. 113. Problems leading to Simultaneous Equations. In Chapter VII. we have had illustrations of problems which were solved by using equations of two unknowns. We now give some further examples on special subjects which ^ere not then considered. SIMULTANEOUS EQUATIONS 167 The number 47 might be written 4 . 10 + 7. What is the sum of the digits of this number ? What number would be formed by reversing the digits ? What is the sum of the number and the reversed number ? What is the sum of the digits of the reversed number ? Ex. 1. — A number has two digits. If 18 is added to it the digits are reversed. The sum of the two numbers is 88. Find the number. Let a;=the units digit and y the tens digit, .". the number =10y + x, and the reversed number =lOx+y. :. 10i/ + a;+18=10x+2/, (1) and 102/ + a;+10x + 2/ = 88. (2) Simplifying (1), 9a;— 9t/= 18 or x — 1/ = 2, (2), lla;+ll2/ = 88 or x + 2/ = 8. Solving z=5, y — 3. .'. the required number is 35. Verification : 35+18 = 53, 35 + 53 = 88. Ex. 2. — If 4 be added to the numerator of a fraction and 3 to the denominator, the fraction becomes |. If 2 had been subtracted from the numerator and 5 from the denominator the result would have been ^. Find the fraction. Let - = the fraction, y x+4 2/+3 1 '' 2 and X- y- -2 -5" 1 ■ 6' 2a:+8 = =2/4 3 and 6x- 12 = =y- -5. 2x-y = = - ■5 and 6a;- y= = 7. Ck>inplete the solution and verify. Sometimes the solution of a problem may be simplified by using some function of x instead of x to represent one of the unknowns. Thus, if two numbers are in the ratio of 7 to 6, we might represent the larger number by x and then the smaller would be fx. 168 ALGEBRA A better way, however, would be to represent the larger by Ix, and then the smaller would be 6.r. By doing so we get rid of the use of fractions. Ex. 3. — The incomes of A and B are in the ratio of 3 to 2, and their expenses in the ratio of 5 to 3. Each saves a year. Find their incomes and expenses. Let S3a; = yl's income, then 82x=B's income. Let $5y = A's expenses, then $3y = B's expenses. .-. 3x-52/ = 400 and 2x-3?/ = 400. Solving X = 800 and y = 400. .-. ^'s income = $3x= $2400 and B's = $1600. .-. ^'s expenses = $5i/= $2000 and B's = $1200. Note. — In solving the problems in the exercise following, the pupil will find that he frequently has the choice of using one, two or more unknowns. Except in special cases, he is advised to use as small a number of unknowns as possible. In each case the results should be verified. BXERCISB 80 1. If 10 men and 4 boys, or 7 men and 10 ^>oys, earn $96 in a day, find a man's wages per day. 2. Two numbers are in the ratio of 5 to 7 and their difiference ia 10. What are the numbers ? 3. The sum of three numbers is 370. The sum of the first two is 70 more than the third, and six times the first is equal to four times the third. Find the numbers. 4. Find three numbers such that the results of adding them two at a time are 29, 33, 36. 5. Divide 429 into three parts so that the quotient of the first by 7, the second by 4 and the third by 2 will all be equal. 6. A workman can save $200 a year. He goes to another town where his wages are 10% greater and his expenses are 5% less, and he can now save $310 a year. What are his wages now ? 7. The denominator of a fraction exceeds the numerator by 3. If 2 is subtracted from each term, the fraction reduces to |. Find the fraction. SIMULTANEOUS EQUATIONS 15§ 8. Divide 120 into three parts, so that J of the first part is greater than the second by 5 and J of the second part is greater than the third by 10. 9. If 6 men and 2 boys earn $56 in 2 days and 7 men and 5 boys earn $57 in 1 J days, how long will it take 3 men and 4 boys to earn $60 ? >» ^10. A number between 10 and 100 is 8 times the sum of its digits, and if 45 be subtracted from it, the digits are reversed. Find the number. ^"V ^^1. The difference of the two digits of a number is 4. The sum of the number and the reversed number is 110. Find the number. 7iJ yf2. The sum of the two digits of a number is 14, and when 18 is added to the number the digits are reversed. Find the number, ^f ^/ When 1 is added to both terms of a fraction the result is \. If 9 had been subtracted from the denominator only the result would have been \. Find the fraction. Ai A number consists of two digits whose sum is 11. If the order oithe digits be reversed, the number thus obtained is greater by 7 than twice the original number. What is the number ? i- ^ p^ The difference between the digits of a number less than 100 is 6. Snow that the difference between the number and the number formed by reversing the digits is always 54. 16. The sum of the reciprocals of two numbers is ^. Six times the reciprocal of the first is greater than five times the reciprocal of the second by \. ' Find the numbers. (The reciprocal of x is -•) X n?/ Divide 150 into two parts such that the quotient obtained by dividing the greater by the less is 3 and the remainder is 2. 18. I wish to obtain 100 lb. of tea worth 34c. per lb. by mixing tea worth 30c. per lb. with tea worth 40c. per lb. How much of each must I take ? 19. Three pounds of tea and 10 of sugar cost $2-40. If tea is increased 10^/ in price and sugar decreased 10%, they would cost $2-52. Find the price of each per lb. 20. Two numbers are in the ratio of 7 to 5. What quotient is obtained when three times their sum is divided by six times their difference I ^ 100 ALGEBRA 21. Show that the sum of any number of two digits and the number formed by reversing the digits is always divisible by 11 and that the difference is divisible by 9. ^2^ A number has three digits, the middle one being 0. If 396 be added the digits are reversed. The difference between the number and five times the sum of the digits is 257. What is the number ? / 23, Divide 126 into four parts, so that if 2 be added to the first, 2~be subtracted from the second, the third be multiplied by 2, and the fourth be divided b}^ 2, the results will all be equal. (Let the result=x.) ^24. There are three numbers such that when each is added to twice t<Ihe^sum of the remaining two the results are 44, 42, 39. Find the numbers. 25? The sum of the three digits of a number is 12. If the units and tens digits be interchanged the number is increased by 36, ariS if the hundreds and units be interchanged it is increased by 198. Find the number. ,- / 26. Find three numbers such that the first with \ of the sum of the other two, the second with J of the others, and the third with \ of tb%others, shall each be 25. 27. A piece of work can be done by A working 6 days and B\2\ days, or by A working 8 days and B 18 days. In what time could each of them complete it alone ? 28. Divide 84 into four parts, so that the first is to the second as 2 to 3, the second to the third as 3 to 4, and the third to the fourth as 4 to 5. ^ y 29. Of what three numbers is it true that the sum of the reciprocals ^oi the first and second is |, of the first and third is \ and of the second and third is J ? 30. Two numbers consist of the same three digits but in inverted order. The sum of the numbers is 1029. The sum of the digits of each is 15 and the difference of the units digits is 5. Find the numbers. .31. A stream flows at 2 miles per hour. A man rows a certain Iflistance up stream in 5 hours and returns in 1| hours. How many miles per hour could he row in still water ? 32. A rancher sold 50 head of horses, part at $125 a head and the balance at §150 a head. After spending $50 he was able to make the first payment of J of the purchase price of 1200 acres of land at $18 per acre. How many horses did he sell at $125 a head ? SIMULTANEOUS EQUATIONS 161 33. A number consists of a units digit and a tenths digit, the units digit being the greater by 1. The sum of the digits is less than twice the number by 2. Find the number. 34. A grocer spent S120 in buying tea at 60c. a lb., and 100 lb. of coffee. He sold the tea at an advance of 25% on cost and the coffee at an advance of 20%. The total selling price was $148. Find the number of lb. of tea purchased.- 35. When 2 is subtracted from each term of a fraction the result is equal to \. Show that the result would have been the same if 1 had been subtracted from the numerator only. 36. It is shown in geometry that the two tangents drawn to a circle from a point are equal. Thus, in the figure AD=AE, etc. If AB=15, BC=U, (7.4 = 13, find x, y and z. 37. If the sides of a triangle are 10, 15 and 19, where will the inscribed circle touch the sides ? (See figure of preceding example.) 38. If A can do a piece of work in m days and B can do it in n days, in what time can they do it working together ? If a; is the number of days required, 111 then - = — |- - X m n m-\-n 39. Use the preceding result to find in what time A and B working together can do a piece of work which could be done by A and B separately in the following number of days : (1) 4 in 10, fi in 15. (2) A in 20, £ in 5. (3) ^ in J, fi in IJ. BXBRCISE 81 (Review of Chapter XII) Solve and verify : /^~x y. 7x-8i/=10, 2. 4a;+72/=-l, -^ 73x+2/= 76, •^ Zx-2y=\Q. ^ 3a;- y = 3. z+732/=147. y 7 4 2 25 32/ + 4x:=2xy. X y X 2y M ALGHBEA /fg) ?±i _ £±1 - 2 ^ y + 2- 2y+l~'^' ^' ""3" - ^T~* 3(a; + y) = 8. /fit 3:r-^=6, .rffc^^-1 ^-3 ^-5 o^ "- 5 "' ap»^ 24-6 4y + ^' = 12. ' x+j,+^=33. ^ x+ y+ z=-3. /^/ 2x+3y- 2 = 5, X + 22/+ 2 = 0, __ \_y^ 3x-4y + 22=l, 3x4- y + 62 = 0. 4x—6y + 5z = 7. y a:+y + 2 = 24. 17 X— 1 3/-f5 _ x+2 ~3 12" ~ ~60"' (x-li)(y-li) = x2/-5. 18. ix-|y+z+l = 3(x-i/) + 52 + 4=x+6i/-22-9 = 0. ^ 2^^102+ix^g 20. x + y = 5, ?/-)-z = 3, 2 + 1^= 1, X — w = 3. 21. 31x4-282/ = 146, 28x4- 31y= 149. (Add and subtract th. equationa and remove common factors.) 22. 97x-592/ = 329, 59x-972/=139. X — 2?y X 4* V 23. What values of x and y will make — —^ and — =-- each equal to x-10 ? 24 Show that x + y + z=\2, 3x4- 4?/ -52= -22, 10x4- 12?/-6z = 4, are indeterminate. Divide unity into two parts so that 18 times the first part mav e^eed 12 times the second by 13. 26, A number of two digits is four times the sum of its digits, and if 18 be added to the number the digits are reversed. What is the number ? O^aT^.^ The tens digit of a number is twice tiio units digit Wh&i> the oumb«r i^ dividend by tbAmjDo oi the digits what muBt thequoUeat be t ^ SIMULTANEOUS EQUATIONS 16S 28. Find a fraction equal to | such that J of the denominator exceeds % of the numerator by 8. /29. ' Two persons who are 30 miles apart are together after 5 hours iiTliey walk in opposite directions, but are not together for 15 hours if they walk in the same direction. What are their rates ? JO. ' A'a age is equal to the combined ages of B and C. Ten years ago A was twice as old as B. Show that 10 years hente A will be twice as old as C. Qi^ A bill of $19-50 was paid in half-dollars and quarters and four times the number of quarters exceeded twice the number of half-dollars by 12. How many of each were used ? f 32j If 5 lb. of tea and 8 lb. of coffee cost $580, and coffee advances lO%''^in price and tea 15% and they now cost $6o3, find the prices per lb. of each before the advance. /dS.y I invest a certain sum at 4% and another sum at 6% and refceive $42 interest. If the sums had been interchanged I would have received $850 more. What were the sums ? If each side of a rectangle is increased by 5 feet the area is increased by 275 square feet. If each side is decreased by 5 feet the area is decreased by 225 square feet. Show that the sides can not be determined from these conditions. 35. Solve ^-^i^^=^ = 4x-^=l. 36. A grocer wishes to mix tea worth 30c. a lb. with tea worth 40c. to make a mixture weighing 601b. worth 36c. a lb. How many lb. of each must he use ? 37. If 3x*— 2a;-f 5 = ax* + 6a; + c, when x=\ or x='2 or x=Z, show thata = 3, 6= -2, c = 5. 38. The tens digit of a number exceeds the units digit by 3. By how much is the number decreased by inverting the digits ? 39. A train is 27 minutes late when it makes its usual trip at 28 miles per hour and is 42 minutes late when it runs at 27 miles per hour. What is the distance ? 40. A piece of work can be done by A working 6 days and B 16 days or by A working 9 days and iS 14 days. How long would it take each alone to do it ? 41. A number has three digits, the units being J of the tens and J of the hundreds, ^f 396 be subtracted the digits a versed. Find the number. y^ v_i Wo^ 184 ALGEBRA 42. When the greater of two numbers is divided by the less the quotient is 5 and the remainder is 2. When 12 times the less is divided by the greater the quotient is 2 and the remainder is 12. Find the numbers. Find four numbers such that when each is added to twice le sum of the remaining three, the results are 46, 43, 41 and 38 respectively. V H|^M^''^*-»'^> 44. If the sum of two numbers is a times the greater and the difference is b times the smaller, show that a — 6 + a6 = 2. CHAPTER XIII GEOMETRICAL REPRESENTATION OF NUMBER 114. Function of x. The value of the expression 3a;— 2 depends upon the vahie of x. Thus, when x= 4, 3, 2, 1, 0, —1,-2, -3, —4, 3x-2=10, 7, 4, 1, -2, -5, -8, -11, -14. When the value of an expression depends upon the value of X, the expression is called a function of x. Thus, 2x— 3, 5x, \x-\-\, are functions of x. What is the value of each of these functions when x = 2, 1, 0, -1, -2? Instead of repeating the words " the expression " or " the function," we might represent the function by a symbol, say y. Thus, if 2/ = 5a;+ 1, when x= 1, 3/ = 6 ; x=3, y= 16. If 7/ = |x+4, what are the values of y when x has the values 6, 3, 0, -1, -8 ? 115. Variables and Constants. A quantity that has nor always the same value is called a variable, while a quantity whose value does not change is called a constant. Thus, the population of a city and tha height of the barometer are variables, while the number of days in a week and the length denoted by an inch are constants. Note. — To do the work of this chapter properly, pupils should be supplied with squared paper. Paper ruled in tenths or eighths of an mch will be found most satisfactory. 165 106 ALOE BRA 116. Connected Variables. Two variables may be so connected that for every change in the value of one there is a corresponding change in the value of the other. Thus, if y = 2x-\-5, for each value of x there is a corresponding value of y. Here x and y are variable quantities, but 5 is a constant. In arts. 20 and 21 we have shown how the changes in two variable quantities may be represented by a diagram. Those diagrams show that for each variation in time there is a corresponding variation in temperature. 117. Graph. A line so drawn as to exhibit the nature of the relation of two variables is called a graph. 118. Arithmetical Graphs. The solution of many problems iji arithmetic might be represented graphically as follows : Ex. 1. — Thb passenger rate on a railway Lg 3 cents per mile. Represent graphically the amount charged for any number of miles from 1 to 10. In the diagram each unit on the hori- zontal line OX represents 1 mile and each unit on the vertical line OY represents 3 cents. The point A shows that the cost for 4 miles is 12 cents. What does the point B show ? The point C ? Read from the figurj the cost for 2 miles, 5 miles, 9 miles. How far can 1 travel for 9 cents, 21 cents, 27 cents T ••; 1 Hj 12 ? / / C / / / / B / / A / / / X No. of Miles Ex. 2. — Represent graphically the simple interest at 2% on $100 for any number of years from 1 to 6. Reading from the diagram (on the next page), what is the interest on $100 in 2 years ? In 5 years ? In 4i years ? What does the point A show ? The point C 1 The point D ? The point half way between C and D ? In how many years is the interest $8, $6, $5, $11, $6-50 7 OEOMETRICAL REPRESENTATION OF NUMBER 167 Place a ruler on the points marked A, B, C, D, E, F. What peculiarity do you notice ? Make a similar diagram, on squared paper, which will give the interest on $200 at 4% for any num- ber of years from 1 to 7. So that your diagram will not occupy too much space vertically, suppose each unit on OY to represent $4 instead of $1. Read from your diagram the in- terest for 3 years, for 5 years. In how many years is the interest $8, $24, $4, $12, $44 ? CJ 8 C 7 Y / / p / / / D / / C / / B / / A / X Time in Years Ex. 3. — A starts at 9 A.M. on a bicycle at 8 miles per hour. He is followed at noon by B on an automobile at 16 miles per hour. When and where will B overtake A ? Each space on the horizontal line represents I hour, and on the vertical line 4 miles. At the end of successive hours A's position will be C, D, etc., and B's will be F, G, etc. The line OP is the graph of A's journey and MQ is the graph of B's. The diagram shows that B overtakes A at the point R, which is 48 miles from the start- ing point and that the time is 3 p.m. How far is A ahead at 12 o'clock ? at 1 ? at 2 ? Solve the problem otherwise and compare the results. Ex. 4. — A starts from P at 9 A.M. to go to Q, a distance of 60 miles, travel- hng at 5 miles per hour. He stops at 12 for one hour for lunch. B starts from ^ at 11 A.M. to go to P, travelling at 15 miles per hour. An accident detains him from 12 to 2. Where and when will they meet ? The graph of A's journey is represented by the upward line drawn from P, and of B's by the downward line drawn from C. The position of each at the end of the auccessive hour? is marked >D the diagram. (See next page.) 07/ P 48 44 40 36 32 28 24 20 16 12 JR } 1 /-> /\ r ( 1 7 1 D, y f C / I 4 / m 9 Id 1.112 ) 2 3 4 5 6 1 108 ALOEBEA They meet at M at about 3.15 p.m. and at a distance from P ol about 26 miles. How far are they apart at the end of each hour from 1 1 to 5 ? Wken did B reach P ? We might solve the problem algebraically. Suppose they are together x hours after 9 a.m. Then A has travelled x— 1 hours at 5 miles per hour, and B X — 4 hours at 15 miles per hour, .-. 5(x-l)+15(x-4) = 60. Solve and compare with the • 8 io 11 12 1 2 a 4 5 resvdts obtained graphically. 119. Graphical Results only Approximate. The last problem illustrates the fact that the results obtained by graphical methods are approximate only. When the problem is solved algebraically we find that they will meet 26 J miles from P at 3.15 P.M. ~ c ^J ?. N ^ \ \ V \ \ ^ ^ \ y r^ Y^ A f^ v^ \ ^ k ^ ^ \ {J ^ l^ \ ^ ^ _ \ BXBRCISB 82 1. A man walks at the rate of 4 miles per hour. Construct a graph to show the distance he walks in any number of hours from 1 to 10. Read from the graph the distance walked in 3, 5, 1\, %\ hours. (Take two units on the horizontal line to represent 1 hour and one unit on the vertical line to represent 2 miles.) 2. In Ex. 1, if he rests 30 minutes after walking each 4 miles, how long will it take him to walk 8, 12, 14, 7, 17 miles ? How far will he have gone in \\, 3^, 5 J, 8 J hours ? 3. A starts running at the rate of 6 yards per second, and 4 seconds later B starts from the same place at 9 yards per second. Construct a graph to show when and where B will overtake A. How far apart are they 6 seconds after A started ? When was A 12 yards ahead oi Bt 4. Oranges sell at 19 cents per dozen. Make a graph from which you can reaxi off the price to the nearest cent of any number from 1 to 12. What is the cost of 2, 5. 7, 8, 10 oranges ? How many can I buy for 3, 5, 8, 13, 16 centa T GEGMETEWAL REPRESENTATION OF NUMBER 169 6. If 8 kilometres equal 5 miles, construct a graph which will enable you to change into miles any number of kilometres up to 20. Read the approximate number of miles in 3, 5, 11, 13, 16, 19, 20 kilometres. 6.* A starts from Toronto at 12 miles per hour to motor to Hamilton, a distance of 40 miles. An hour and a half later B starts from Hamilton to drive to Toronto at 8 miles per hour. By means of a graph, find when and where they will meet. 7. The distance from A to B is 10 miles, 5 to C 8, C to Z> 8, i) to E 10 miles. A mail train, which leaves A at 10 a.m., arrives at B at 10.24, C at 10.48, D at 11.12, E at 11.40. An express train leaves E at 10.24 and without stopping reaches A at 11.28. If the mail train stops 4 minutes at each station, show graphically : (a) when and at what point they pass each other, (6) how far they are apart at 10.30 and at 11.12, (c) when the express passes through B. 120. The Axes. In the diagram the line OX is called the axis of x and Y the axis of y. We will call the measurement along OX, X, and along OY, y. For the point A the x measure- ment is 3, and the y measurement is 1. What are the x and y measurements for the points B, C, D, El. Examine the x and y measure- ments for each point marked on the line OP. What equation con- nects the values of x and y for each point on the line OP ? For each point on OQ, OR, OSl OP is the graph of the equation y=2x, OQ of i/=4x, OR of y=x, and OS of y'=lx. The X and y measurements of every point on the Une OP satisfy the equation y—2x. This equation is not satisfied -7- V / y / t 6 U k i / E / ■5 / / / / - 4 / / r /B / 3 / / D / % 2 / "c '^ "* ' A ^ A ?: < ^ 1 ^ 1 2 i \ 4 ? 6| 170 ALGEBRA by the x and y of any point not on the line OP. by the x and y of the points A, C, D, E 1 Is it satisfied 121. Equation of a Line. Since the equation y—2x is satisfied by the values of x and y for each point on the line OP and by no other points, the equation y=2x is called the equation of the line OP. What is the equation of OQ ? of Oi? ? of OS ? 122. To Construct the Graph of a given Equation, Ex. — Construct the graph of y=lx. Here when y j ^^ ,^ ^ .^ 7^ r , ■' ^^ .^'- Ll >C x = 4, 2/ = 2, a; = 6, j/ = 3, etc. To find the point where x = 2, 3/=l, count 2 units from along OX and then 1 unit upwards. Find in a similar manner the points where x = 4, y = 2; x=&, y=3 ; x=8, j/ = 4 ; x=10, y = 5. Join all the points located. They are all seen to lie on the same straight line passing througli 0. This lino is the graph of the equation ;/ = ^x. 123. The Origin. All the lines we have so far considered have been drawn through the point 0. This point is called the origin. The X and y measurements of the origin are x=0, ?/=0. These values satisfy the equation y=|.r, and consequently the graph of this equation should pass through the origin as the figure shows. EXBROISB 83 Construct the graphs of : 1. y=x. 2. y—5x. 3. y=\x. 4. y—^x. 5. ^y—x. 6. 5y=2x. 124. Coordinates. In the diagram is the origin, XOX' is the axis of x and YO Y' is the axis of y. The x measurement of the point A is 4, and the y measurement is 2. GEOMETRICAL REPRESENTATION OF NUMBER 171 y Q 1 r rn D R A X -i^^^ X - Q ^ P - - - r 1 ,r^ L^ .1_ _ _ These are called the coordinates of the point A, 4 being the ,r coordinate and 2 the y coordinate. The coordinates of the point A are written (4, 2), the X coordinate being written first. Similarly, the coordinates of B are (3, 5) and of C (6, 7). So far all our measurements have been made from towards the right and then upwards. We might also measure from to the left and downwards. When we do so we indicate the change in direction by a change in algebraic sign. Thus, to reach the point D we ixieasure 2 units to the left and then 3 units upwards. Therefore the coordinates of D are ( — 2, 3). Similarly, the coordinates of £■ are ( - 6, 5), of i*^ ( — 3, — 3), of O (3, — 5), and of H (0, -4). What are the coordinates of P, Q, R, S, 1 Mark on the diagram the points whose coordinates are (2, 2), ( — 4, 6), (2, -5), (-1, -3), (0, 4), (-3, 0), (0, -3). Using squared paper, take the origin at the intersection of two lines. Mark the points which would be indicated thus : (2, —3), (5, 6), ( — 3, 7), ( — 6, —2), (3, 0), (0, 3). Mark any other four points and show how their positions would be indicated. 125. Quadrants. That part of the plane between OX and OY is called the first quadrant, between OY and OX' the second quadrant, between OX' and OY' the third and between OY' and OX the fourth. Thus, the points A and B are in the first quadrant, D and E in the second, F and S in the third and P and in the fourth. In which quadrant are both x and y negative ? 126. Plotting Points. When we represent the position of a point with respect to the axes XOX' and YOY', we are said to plot the point. When two points are plotted the distance between them may be obtained by adjusting the points of the compasses 172 ALGEBRA to tlic two points and transferring the compasses to the line OXy or any other line, and reading off the distance. Plot the points (3, 5) and (6, 1), and see if the distance between them is 5. BXBRCISEI 84 1. In what quadrants are the points (3, 4), (4, —1), (—5, 3), (-1, -2)? 2. Plot the points (1, 2), (4, -6), (-3, 7), (-5, -2). 3. Plot the points (5, 0) and ( — 3,0). What is the distance between them ? 4. Where are the points (0, 0), (0, 2), (-5, 0), (4, 0) situated ? 5.* What is the distance between the points (6, 4), (1, —8) ? 0, What kind of figure is formed by joining the points (0, 0), (4, 0), (4, 4), (0, 4) in order ? What is its area ? 7. What kind of a triangle is formed by joining the points (0, 2), (2, 6), (2, 2) ? What is its area ? 8. Plot the points (1, 1), (1, 3), (2, 1), (3, 3), (3, 1). Join them in order. What letter is formed ? 9. What is the area of the figure formed by joining (1, —3), (-5, -3), (-5, 6), (1, 6) in order ? 10. The angular points of a triangle are (6, 0), (3, 4), ( — 2, 0). Construct the triangle and find its area. Measure or calculate the lengths of the sides. 11. What is the length of the perpendicular from the point (5, 8) to the line joining (3, 2) and (7, 2) ? 127. Complete Graphs. Y / y y X (6, 3) y ^ (4, 2) X )< ^ (2, 1) X ^ > r^ ^(1 2, ■1) ^ -r- *,• 2) y; 4 6,- 3; Y ' In art. 122 we constructed the graph of the equation y=\x^ but only for positive values of X and y. The diagram, which is here repeated, shows that the line ako passes through th© points GEOMETRICAL REPRESENTATION OF NUMBER 173 (—2,-1), (—4,-2) and (—6,-3). This is as we would expect because x=—2,y= — l; x=—4:,y=—2; x= — 6, y= — 3, all satisfy the equation y^=^x. 128. Linear Equation. It is seen that the graphs of all the equations so far constructed have been straight lines. This is true concerning all equations of the first degree. For this reason an equation of the first degree is sometimes called a linear equation. Since a straight fine is fixed or determined when any two points on it are fixed, it follows that to construct the graph of an equation of the first degree, we need to determine only two points on it. 129. Lines not passing through the Origin. Every equation of the form y=mz represents a straight line passing through the origin, because the equation is satisfied by x=0, y=0. If the equation contains a term independent of x and y, it represents a straight line which does not pass through the origin. Thus, 2/ = 2x4- 1 represents a straight line which does not pass through the origin, because this equation is not satisfied by x = 0, y = 0. Ex. 1. — Construct the graph of y==2x-\-\. The coordinates of two points on the line are x=0, y = \ and a;=l, 2/ = 3. Locate these two points and draw the un- limited straight line whicli joins them. This is the required graph. The diagram shows that it also passes through the points (2, 5), (—1, —1), (-2, —3), (-3, — 5). Do the coordinates of these points satisfy the equation ? In constructing the graph of an equa- tion by locating two points on it, the pupil should try and determine two points whose coordinates are inteaers. Y / 1 (i 3) X J ,1) X / w7 -1 ^ ; ) / f L ^ 174 A LOB BRA Ex. 2. — Construct the graph of 3x+4y=15. „ 15-3x J , /x=l, w = 3 Here y = — - — and when - ^ Plot the points (1, 3) and (5, 0) and join them giving the required graph. We might have found the points at which the graph cuts the axes. Thus, when x = 0, 2/ = 3J and when 2/ = 0, x = 5. The required line is then found by joining the points (0, 3|) and (5, 0). If the latter method is followed and fractions appear in the coordinates of either of the points found, the unit of measurement should ijo changed, in this case, by taking /ottr spaces as the unit instead of 07ie. When the unit is not one space, it should be clearly shown on the diagram what the selected unit is EIXBROISE 86 1. Find two pairs of values of x and y which satisfy x-{-y=6. Plot the points whose coordinates are the values found and construct the graph of the equation z-\-y-=6. 2. What are the coordinates of the points at which the graph in Ex. 1 cuts the axis of x, the axis of y 1 Construct the graphs of the following equations : 3. y=x+3. 4. y=x—3. 5. y=2x-3. 6. y=3x—2. 7. x+2y=l. 8. x-2y=l. 9. 2x + 3y=\2. 10. 3x-4y=\6. 11. 5x+6y=n. 12. Construct the graph which cuts off 4 units from the axis of x and 6 units from the axis of y. Find the area of the triangle which this line forms with the axes. 13. On the same sheet construct the graphs of x—y=^lO and i; 4- 2^=7. Wliat are the coordinates of the point at which they /ntersect ? Do the coordmates of this point satisfy both equations ? GEOMETRICAL REPRESENTATION OF NUMBER 175 14, Will the point (3, 4) lie on the graph of the equation 4x+3?/=24 ? Which of the following points lie on it: (2, 6), (0, 8), (6, 0), (9, -4), (5, 2), ( — 1, 9) ? V^erify by constructing the graph. 15, By constructing the graph of 2a;-f 3t/=24, find three sets of positive integral values of x and y which satisfy this equation, 16, Why is there an unlimited number of positive integral values of X and y which will satisfy 2x—3y=2'i, but only a limited number which wiU satisfy 2a; +3?/ =24 ? 130. Graphical Solution of Simultaneous Equations. In this diagram are shown the graphs of the equations x+y = 5 and 2x—3y=lo. The coordinates of the point P, at which the lines intersect, must satisfy both equations. The coordinates of P are (6,-1). .*. x=6, ?/= — 1, must be the values of X and y which satisfy both equa- tions. We have therefore obtained the sohition of these two equations Graphically. Since it is evident that two straight lines can inter.sect at only one point, it must follow that there is only one pair of roots of tivo simultaneous equations of the first degree. In this diagram are shown the graphs of (a) x-y=3, (6) 2x-y=7, (c) iJa;+?/=-7. At what point do the graphs of (a) and (6) intersect ? (a) and (c) ? (6) and (c) ? Is there any point which is common to the three lines ? Are there any values of X and y which will satisfy these three equations at the same time ? When three equations in x and y are all satisfied by the same values of x and y, what peculiarity will appear in their graphs ? \ / \ s , \ X \ <$. ^ X f s \ k V R K ^ \ y 1$, ,^ ^ 3 h^\ ^ y > }{ -c ■iV \f 1 ■Jo 1 Vj ^ / / ^A r X -ij \ A X 1 \ \ / \ / \ \ / f\ A / \ /, / \ J V y \ I ■i- J \ / 1 \ f -L »^ L 176 ALQEBHA "s V, y •< L '' [f N ^ fP *>. ■v. f 'V. <, ^ M \P > ( 1 s *V -^ ^v^l \ > *N X *< "s s ^ y V k 131. Special Forms of Equations. ( 1 ) In this diagram are drawn the graphs of a;4-2t/=6 and x+2?/=2. The lines which these equa- tions represent are seen to be parallel, that is, there is no point at which they intersect. This in equivalent to saying that these equations have no solution. Compare with art. Ill, where inconsistent equations were discussed. (2) If we draw the graphs of x—3y=l0 and 2x—6y=20 on the same sheet, we shall find that they represent the same straight line, so that any points which lie on the graph of one of them will also He on the graph of the other. These equations are indeterminate (art. 111). (3) An equation like x=Z may be written a;+Ch/=3. This equation is satisfied by a; =3 and any value of y. Thus, x=3, y=l ; x^3, y=2 ; x=S, y^lO, etc., satisfy the equation. If we plot the points which have these co- ordinates, we see that x—3 represents a straight line parallel to the axis of y and at a distance 3 to the right of it. Similarly, x= —3 represents a line parallel to the axis of y and at a distance 3 to the left of it. Also y= —4 represents a line parallel to the axis of x and at a distance 4 below it. Thus an equation which contains only one of the variables X and y represents a line parallel to one of the axes. What line does x=Q represent ? 2/=0 ? EXERCISE 86 (1-4. Oral) 1. What is the graph of x=3 ? Of a;-l-3=0 ? 2. What is the graph of i/-4=0 ? Of 2y+3=0 ? 3. If x-\-Zy=\\, express x as a function of y and y aa & function of X. GEOMETRIC AL EEPRESEN^^TION OF NUMBER 177 4. If 3x—2y=6, express x as a function of y and y as a function of X. Solve graphically and verify : 5. x+'Sy=9, 6. x+ y= 8, 7. x— 2j/= 6, 2x+ y^8. 3x—iy=lO. 2x— 3t/=ll. 8. 2z+3y= 6, 9, 2?/== x, 10. x+iy=9, 3x-\-2y=14:. 10jr=4x-2. 3x-8i/=-3. 11. Show by graphs that the equations x-\-y=5, 2x-r3y— 12, 3x—2y=5 have a common pair of roots and find them. 12. Show graphically that 2x+3y=13 and \x-\-%y=2 are incon- sistent. What is peculiar about the graphs of these equations ? 13. Show graphically that no values of x and y will satisfy all of the equations x-{-y=4:, 2x—y=ll, 4x+2/=13. What values satisfy ••.he first and second, the first and third, the second and third ? 14. Show the coordinates of the points where the graph of y=2x-\-3 cuts (1) the axis of x, (2) the axis of y, (3) the graph of ?/=6— x. 15. Show by graphs that the values of x and y which satisfy 2x— 32/4-1=0 and 5x-2?/— 14=0 will also satisfy 3x—4:y=0 and x—2y+2=0. BXEIRCISB 87 (Review of Chapter XUI) 1. At what point does the graph of x-{-y = 5 cut the axis of x 7 The axis of y ? Construct the graph. In the same way construct the graph of x + 4!/= —4. At what point do they intersect ? 2. How does it appear geometrically that two equations of the first degree can have only one set of roots ? 3.* Plot the points (0, 0), (-3, 4), (3, 12), (-2, 0). What is the distance between each conscoutive pair of these points ? 4. From a certain point a man walks 5 miles E., then 4N.; than 2W., then 3N., then 3E., then 4S. Using squared paper, determine by measurement how far he is now from the starting point. 5. A man walks 8 miles W. and then 5S. Find by calculatioo how far he must now walk to reach a point 4 miles E. of his starting point. N 179 ALOlCBiiA 6. If 11 lb. equal 5 kilogrammes, make a graph from which you can express any number of kilogrammes in lb. or lb. in kilogrammes. Read from the graph 3J kilogrammes in lb. and 8J lb. in kilogrammes. 7. What is the perimeter and area of the triangle whose angular points are (0, 0), (5, 0), (0, 12) ? 8. How do you show that the point (3, —2) lies on the graph, of 5x — 2y=l9 1 Which of the following lie on it: (6, 5), (1, — 7)i (- 3, -17), (4, 1), (-2, -12), (5, 3) ? 9. Find the area of the triangle formed by joining the points (5, 9), (8. -6). (-7. -6). 10. Draw the triangle whose vertices are (2, 0), (10, 0), (5, 6) and find its area. Why do the points (2, 0), (10, 0), (8, 6) determine a triangle of the same area ? Solve graphically and verify : 11. x+2y=l2, 12. 3x-'iy= 0, 16. t/-x = 4, x~2j= 2. 4x-3i/ = U. x='2. 14. y-2x=-3, 15. 2x+7y = 52, 16. j,' = Jx + 4. x+2y=li. 3x-5!/=16. !/ = Jx+5. 17. What is the area of the figure formed by the lines whose equations are: x = 4, x=— 2, y = 3, y= — l ? 18. What are the coordinates of the middle point of the line joining the points (2, 3) and (6, 5) ? 19. On the same sheet draw the graphs of the equations y = x+4, 2/ = 4x-2, y = 2x+2. What peculiarity is presented by the graphs ? What conclusion do you draw concerning these equations ? 20. Draw the graphs of 2x-f-3y = 20, 4x-|-6!/ = 35 on the same sheet. What do you conclude as to the "solution of these equations ? Determine graphically whether these sets of equations are consistent or inconsistent : 21. X- y= 4, 22. x+2j/=10, 23 -x+ y= 8, 4x4- 2/ = 26, 3x- y= 9, 3x4-2?/= 13, 2x-5y= 2. 2x- y= 1. 5x-3i/= 9. 24. Describe the crianglt .vhose sides are represented by the equations: 3x-r2y=14, ox— %= — 14, x-!- 102/= — 14. W^hat are the coordinates of its vertices 7 (Verify by solving the equatiorw •*> oairs.) GEOMETRICAL REPRESENTATION OF NUMBER 179 25. At what point do the graphs of 2x+Zy=\2, 3x — 2y = 5 intersect ? At what angle do they seem to intersect ? 26. A teacher's salary is increased by $50 each year. His salary for the first year is $750. Construct a graph from which you can read off his salary for any year. What is his salary for the 8th year ? In what year would his salary be $1300 ? 27. In the process of solving 2x — 3y=l, 3x + 22/ = 8, by eliminating y we have 2x-3y=l, I 4:x-&y= 2, I x=2, j x = 2, 3x+2y = S. I 9x + 62/ = 24. j 2a;-3y=l. | y=l. On the same sheet show the graphs of each of these sets of equations, and thus show that they all determine the same point and that the four sets are therefore equivalent 9 t CHAPTER XIV HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE 132. In Chapter IX. we defined the terms highest common factor and lowest common multiple, and showed how they were found in simple cases. When the expressions under consideration can be factored, the H.C.F. and L.C.M. can at once be written down from the factored results. A few examples are here given of a more difficult character than those previously considered. Ex. 1.— Find the H.C.F. and L.C.M. of x^y-{-lxy^-\-12y^ and x^y—x^y^-~l2xy^. x*y-\-lxy*+l2y^ =y{x*^lxy+\2y^) = y{x^-'^y){x + ^y). x'y — x*y*— \2xy^ = xy(x*~xy~ l2y*) = xy{x — 4y){x-\-Sy). Here the common factors are y and x+Zy, and since the H.C.F. is the product of all the common factors, .-. the H.C.F.=2/(x+3!/). The L.C.M. is the expression with the lowest number of factors which will include all the factors of each expression, .-. the L..C.M. = xy{x + 'iy){x + 2y)(x-iy). Ex. 2.— Find the L.C.M. of a;2— 1, x^+l, x*—x and x^+x^+L x*-l = (x+l)(x-]). x^+l = {x+l){x*-x+l). x*—x=x{x'— l) = x(x—\)(x*-{-x-\- 1). x* + x*+\ = {x*+l)*-x' = {x* + x+l){x*-x+l), :. the L.C.M. = a;(x+l)(a;-l)(a;» + a;-|-l){x«-x+l). H.C.F. AND L.C.M. 181 If the multiplications be performed the L.C.M. will be found to be x' — x. It is customary, however, to leave the result in the factored form, as it is in this form that it is usually made use of. BXBRCISB 88 Find the H.C.F. and L.G.M. of : 1.* ^x'^y^z, 8xy*z% \2axyH. 2. x^—y^, xy—y'^, x'^'—xy. 3. a^~h\ o6+62, a'^+2ab-\-b^. 4. x^-lx-\-\2, x^-\-2x-\5, a;2-9. ^. a2+8a+15, a^-2a-25, a'^^Za~\Q. Jo. 3x2- 12x4- 12, 3x2-12, 3x2 ^3x-6. [/I. x'^—xy-\-xz—yz, xy—y"^. ^. m3-8, m*n^-47n^h^, 4»i2-16ffi+16. j^. 6a3-663, 2a^+2a^b+2abK f^. a^+ab-ac, a^-i-b^-c^+2ab. [kU. a2-62-c2-26c, b^-c'-a^-2ca, c^-a^-b^-2ab. ^12. x3+2/3, x*+x^y^+y*. - )^ 3x2+7x-6, 3x2-llx+6, 6x2-13x+6. 14. 10ax-2a+15cx-3c, 25x2-1, 25x2-10x+l. 15. x3-5x2+6x, x3-3x2+5x-15. j(^16. u*—v*, u^—v^, u^—v\ u—v. ±1^ xH2x2-8x— 16, x3+3x2-8x-24. 18. Show that the product of x*— 8x+15 and x^-\-x—\2 is equal to the product of their H.C.F. and L.C.M. 19. The L.C.M. of a2-5a+6 and a'^ -6 is a3-3a2_4a4-i2. Supply the missing term. 20. Find two trinomials whose H.C.F. is x—2y and whose L.C.M, is z'— 7xy'+6v*. 182 ALGEBRA Ex. 1.— Find the H.C.F. and L.C.M. of a;2-|-2.r-3 and .r^-S.r+S. Here ar* + 2x— 3 is readily factored, but none of the methods <reviously given will apply in factoring x*— 8x + 3, except by using tlie factor theorem of art. 101. The difficulty is, however, easily overcome thus : x» + 2x-3 = (x-l)(x+3). If the expressions have a common factor it must evidently be either X— 1 or x+3. By using the factor theorem, find if x— 1 or x + 3 is a factor of x»-8x+3. When x-l=0 or x= 1, x3-8x-(-3= 1 -8-f3= - 4, .". X— 1 is not a factor. When x+3-0 or x=-3, x»-8x + 3= -27 + 24 + 3 = 0, x+3 is a factor. How can we obtain the other factor of x* — 8x+3 ? We now have x'' + 2x-3 = (x- l){x + 3), and x*-8x + 3 = (x+3){x*-3x+l). /. the H.C.F. = x-l-3, and the L.C.M. = (x+3){x- l)(x2-3x+i), Ex. 2.— Find the H.C.F. and L.C.M. of a;2_7a;^10 and a:^— 6a;2+lla;-6. The factors of x*-7x+10 are (x-5)(x-2). Here it is evident that x— 5 is not a factor of the second expression, since its last term is — 6, which is not divisible by 5. Is X— 2 a factor of x^ — 6x'+llx — 6 ? Complete the solution. SiXBROISB 89 Find the H.C.F. and L.C.M. of : ^* x2-3x+2, xs-6x2-f 8x-3. 2. a'-6a+5, a^-lQa^-f I7a+1. 3. x8-2x2+4a:-8, 2xS-7x2+12. /f. a»-o2+a-l, 3a3-2a2+5a-6. yt a:8+3a;2_4a;. x3-7x+6 LC-.F. AND L.C.M. VSa 6^^If X — 2 is a common factor of xHSa:^— 9a;— 2 and x^— 4x2+3x+2, find their L.C.M. yf. Reduce to lowest terms : a2— 3a6+262 x^—2x^—'ix and a3-19a62+3063 2x*—Ux^—12x 8. Find two expressions of the third degree in x, whose H.C.F. is a;2_5a;_j_6 and whose L.C.M. is x*— lOx^+Soz^— oOx+24. 133. Method of finding the H.C.F. of two expressions which can not be factored by the usual methods. From the preceding it is seen that Lhe chief difficulty in finding the H.C.F. of two expressions is in factoring the given expressions. If neither of the expressions can be factored by the usual methods, another method may be used which depends upon the same principle as that of finding the G.C.M. of two numbers in arithmetic. 134. Fundamental Theorem. This method of finding H.C.F. depends upon the following theorem : If 0^ is a common factor of any two quantities, then oc is also a factor of the| sum or difference of any multiples of those quantities. Thus, X is a common factor of mx and nx. Then mx~nx, mx—nx, pmx-\-qnx, rmx — snx, are each the sum or difference of multiples of mx and nx. It is evident that each of these is divisible by x, the quotient in each case being found by division, thus : x)mx-\-nx x^mx — nx m —n x)pmx + qnx pm -\-qn x)rmx — snx m -\-n rm —an The way in which the theorem is applied is shown in the following examples. Ex. 1.— Find the H.C.F. of x3-f4x2+4a:+3 and a;3+3a;2+4x+12» Any common factor of these is a factor of their difference, which is x*-9. But x*-9 = (x-3)(x+3). .-, the H.C.F. is x-3 or x + 3 or (x-3)(x + 3). 184 ALQEBRA It is evident that x— 3 is not a factor of either expression, since their terms are all positive. Therefore if they have a common factor it must be x+3. By applying the factor theorem, or by division, we find that x+3 is a factor of each, and since it is the only common factor, it must be ♦he H.C.F. Ex. 2.— Find the H.C.F. of 3a;3-17x2-5a;+10 and 3a;3-23a:2+23a:-6. TheirdifEerence = 6x*-28x+16 = 2(a;-4)(3x-2). Now 2 is not a factor of either and may be discard_ed, also x — 4 is not a factor, since 4 is not a factor of 10 nor of 6. Therefore if there is a common factor it must be 3x— 2. Divide 3x— 2 into one of them and see if it divides evenly. If it does not there is no common factor but unity. If it does divide evenly into one of them, it is not necessary to divide it into the other, for if it is a factor of one of them and also of their difference it must be a factor of the other. Ex. 3.— Find the H.C.F. of 3x3-13a;2+23a;-21 and Qx^-\-x^-4Ax+2\. Multiply the first by 2 and subtract the product from the second and we get 27x»-90x + 63 = 9(x-l)(3x-7). Now since 9(x— l)(3x— 7) is the difference of two multiples of the given expressions, it must contain all their common factors. Which of these factors may be discarded ? Complete the solution. We might have obtained the H.C.F. thus : The sum of the expressions is 9x»-12x*-21x = 3x(x+l)(3x-7). This expression contains all the common factors of the given expressions. Complete the solution by this method. The object in each case is to obtain from the given expressions an expression of the second degree. If this expression 'can not be factored, it must be the H.C.F., if there is any common factor other than unity. If it can be factored the H.C.F. can then be found either by the factor theorem or by ordinary division. H.C.F. AND L.C.M. 185 In obtaining the expression of the second degree, the last problem shows that it is sometimes easier to eliminate the last terms than the first terms. Ex. 4.— Find the H.C.F. and L.C.M. of 6x3— 5a;2— 8a;+3 and 4^3— 8a;2+a:+3. Eliminate the absolute terms and show that 2x — 3 is the H.C.F. Since 2x — 3 is a factor of each, the other factors may be found by division, then 6a;»-5a;»-8a;+3 = ((2a;-3)(3x* + 2a;-l), 4a;»-8x*+x+3 = (2a:-3)(2a;»-a;- 1), .-. the L.C.M. = (2a;-3)(3a;» + 2a;-l)(2a;»-a;-l). Why is it unnecessary to factor 3x* + 2x— 1 and 2x* — x— 1 7 Ex. 5.— Find the H.C.F. of a;*— 4a:3+10a;2— llx+10, (1) and a;*— a;3— 4a;2+19a:— 15. (2) Subtract (1) from (2), and we get 3a;»-14a;» + 30a;-25. (3) Multiply (1) by 3 and (2) by 2 and add to eliminate the absolute terms. Remove the factor x and we obtain 5x3-14x» + 22a; + 5. (4) The common factor we are seeking must be a factor of both (3) and (4). Eliminate the absolute terms from (3) and (4) and show that the H.C.F. is x»-3x+5. Find also the L.C.M. Ex. 6.— Find the H.C.F. of 8a:*+4a;3+4a:2— 4a; and 6a;*+2a;3+2a;2— 4a;. Here 4x is a factor of the first expression and 2x of the second, and therefore 3a; is a common factor. Remove these sioEiple factors and find the H.C.F. of the quotients, and show that the H.C.F. is ^x(x» + a;+l). 135. Product of the H.C.F. and L.C.M. Suppose that x is the H.C.F. of mx and nx, so that m and n have no common factor. 186 ALGEBRA Then tlie L.C.M. of mx and nx is mnx. But xxmnx= mx x nx, therefore the jiroducl of any two quantities is equal to the product of their H.C.F. and L.C.M. Is a similar theorem true concerning any three quantities mx, nx and px ? If the H.C.F. of two quantities has been found, we might therefore find their L.C.M. by dividing their product by the H.C.F. BZBROISB 90 Find the H.C.F. of: @l* x3-7x2 + 13x-15, x3-6a;2+x+20. §. a3-10a2+33a-36, a^-2a--2'ia+m. . 6x3 + 10a;2+8x+4, 6x3—2x2-4. 1/4. 2x3-5x2-20x+9, 2x3+x2-43x— 9. t^ 263 + 562-86-1.5, 463-462-96+5. ^. *3x3 + 17x2j/-44xi/2_28)/3, 6x3-5x2^-33x?/2+28j/'. ij^ 2a3-3a2-4a+4, 3a*-4a3-10a+4. ^. 2x^-12x3 + 19x2 -6x+ 9, 4x3- 18x2+ 19x-3. 9. 18^55_3a4ft_i2a36_3a26, 12a5c-6a*c-9a3c+3a2c. 10. x3— x2-2x+2, X*— 3x3+2x2+x— 1. Find the L.C.M. of : A.1. x3-7x-6, x3-4x2+4x— 3.' j^2. x3+6x2+llx+6, x3+7x2+14x+8, x3+8x2+19x+12. /3. 2x3+9x2+7x— 3, 3x3+5x2- 15x+4. ^4. x3-6x2 + llx-6, x3— 7x2+14x— 8. ]jCb. 20x*+x2-l, 25x*-10x2 + l, 25x*+5x3-x-l. 16. Find a value of x which will make 'x3 — 13x+12 and x'— 6x2— x+30 ^acti equal to 0. 17. The L.C.M. of two numbers is 70 and the H.C.F. ta 7. If ooa of the numbers is 14, find the other H.C.F. AND L.C.M. 187 18. The H.C.F. of two expressions is x— 2, the L.C.M. is x^— 39x+70. If one of the expressions is x^—lx+10, find the other. 19. Two integers differ by 11. If they have a common factor, other than unity, what must it be ? EXERCISE 91 (Review of Chapter XIV) Find the H.C.F. and L.C.M. of : jt!*' x*-20x+99, x*-24x+143, a;*-21a;+110. itf a;*-15x+36, x'-27, x3-3x«-2a;-f 6. . ''■- • ^. a*-b*,a*-2ab + b*,a^-b\ ' A. x3-2x*-15x, x^ + x*— 14X-24. y<^ 4a'-12a«-a + 3, 2a3 + a*-18a-9. 6. X* — ax— 6x-f-a6, x^ — bx—cx-\-bc. 7. x» — 6x«+llx-6, x* + 4x« + x-6. 8. x« + 3x« + 3x* + 5x-12, x*-4x'- 19x«+ IOx+ 12. ^ 2a«+15a' + 39a« + 40a+12, 2a« + 9a»-2a*-39a- 18. fin. x*—6x*2/+13x«2/*- 12x1/3 + 42/*, x* + 2x3j/ — 3x*i/* — 4xt/3 + 4V*. ' </ll. x* + x*2/* + 2/S X*— 2x3?/ + 3x2^ = — 2x2/' + y*- 12. Show that two consecutive integers can have no integral common factor except unity. 13. Two odd integers which differ by 6 have a common factor other than unity. What must it be ? 14. Find the H.C.F. of x^ + a' and x^ + x*a* + a^. 15. If the H.C.F. of a and b is d, show that the L.C.M. is -=- • a 16. If a is the H.C.F. and b is the L.C.M. of three quantities, show that the product of the quantities is a^b. 17. For what common values of x will x' — 3x*-x+3 and x*-4x»+12x— 9 both vanish ? 18. Find two expressions of the second degree in x, whose H.C.F. is x-1 and L.C.M. is x^-Sx-^ + 17x— 10. 10, D J 18x»-3x« + 2x+8 ^ , , , ^^ 19 Reduce -,. . , ^ , — - — — ^^ to lowest terms. ^pr 12x»+8x* — 7x+12 ^'' CHAPTER XV FRACTIONS In Chapter IX. fractions were introduced, and simple examples of operations upon fractions. ^ In this Chapter the subject is extended and applications made to more complicated forms. 136. Changes in the Form of a Fraction. Both terms of a fraction may be multiplied or divided by the same quantity without altering the value of the fraction. As previously stated, the only exception to this rule is, that the quantity by which we multiply or divide must not be zero. The rule might be stated in the symbolic form : a _ ma na _ a b ~~ mb nb ~~ 6 The case in which the terms are multiplied or divided by —1 deserves special attention. From the rule of signs for division — r is seen to be the a , . —a same as — - , so also is — j— • —a a a •• b ~ -b~ b — fl d — d Similarly, _& = fc = 6 = a -b It is thus seen, that the vahie of a fraction is not changed by changing the signs of both of its terms ; or by changing the sign of one of its terms and at the same time changing the sign before the fraction. FRACTIONS 189 Since (a— 6) /;( — !) = — a-f-ft or h — a, it is seen that a—b and h—a differ only in sign, or that each one is equal to the other multipHed by — 1. That is, a—h = — {h—a) and b—a= — {a—b). a—b _ (g— 6)x( — 1) _ b—a _ a—b _ b—a c — d ~ (c— d)x( — 1) ~~ d—c ~ d—c ~ c—d' Also, since ( — a) x( — 6) = ( + a) x ( + 6) = a6, it follows that (o — 6)(c — d) = {6 — a)(d— c), m m — m m (a-6)(c-d) {b-a){d-c) (a-6)(d-c) (6-o)(c-d)' (a-a:)(b-y) ^ ( x-a){y-b) ^ ( a-x){y-b) ^ ^^^ {b—a){a—y) (x—b){y—a) (b—x){y—a) E3XERCISE! 92 (1-29, Oral) Express these fractions in their simplest forms with no negative signs in either term : -2 4 -6 -3a 4 — 2 —9 — a —a 5 — 4o6 — ax 2/ —m 2b — bx — 3x— 5 —axb —a.—b —x.—y 9. = 10. J-- 11. ^" ^ 7 —b —c Express with the numerator a — b : b — a b — a b — a 13. — :r • 14. — — r • 15. —3 —0 x—y Express with the denominator c—d : — 5 — x. —y a — 6 17. :i 18. —J -■ 19. :i a — c a — c a — c Express with the positive sign before the fraction : -4 4 a 21.—-=-- 22.-—;:. 23. -T- 7 —7 6 X a+b x—2 c—d 25. — ; • 26. — , • 27. — — • 28. — ——, a—b a—b x—y c+o 29. What is the relation between a , a a+b , b+a b—a a — 6 and . ; and r — . — ^ and — s— ? ic—y y—x a—b b—a —3 3 —a. —b 16. b-a -2{c-d) 20. -m{x-y) d- 24. a—h c 190 ALGEBRA rrr • (? — ?){? — '■) 30. Write . .. _ . in four equivalent forms, with the positive sign before the fraction. 31, Which of the following are equal in value : (a-b){b-c){c-a), {b-a){c-b){a-c), {a-b){c-b){a-c), (a—b){b—c){a—c), (b—a){b—c)(a-c) ? 137. Reduction of Fractions to Lowest Terms. The formula ax a ,— = 7 may be used to reduce a fraction to its lowest terms. ox ^ by dividing both terms by all the common factors. Ex. 1.— Reduce ^ "^^ x' + y^ = (x-\-y)(x^~xy+y*). x* + x*y* + y* = {x* + y*)* -x^y^ = {x' + xy + y*}{x*-xy + y*), .'. the fraction = ,— — - • x^+xy+y* Ex. 2.-Reduce „._,.^^,^,^ • a» + 6*-c» + 2a6 = (a + fe;*-c* = (a + 6 + c)(a4-6-c). Complete the reduction. 1- o T. J x2-lla:+28 Ex. 3.-Reduce ,^3_6,2+7,_60 " a;»-llx+28 = (x-4)(x-7). Which of these factors can not divide into the denominator 7 Complete the reduction. T. . T^ J 3x3- 15x2- 19a;+6 Ex. ^.-Reduce e.3+3.2-5.+ l " Here the factors of neither term can be readily obtained, 30 the common factor must be found by the method of art. 134. Eliminate the x' and we obtain 33x* + 33x-ll or 1 l(3x« + 3x- 1). This expression must contain any common factor of both terms. Since 3x* + 3z— 1 can not be factored, what conclusion can be drawn ? Complete the reduction. FRACTIONS 191 EXBRCISB 98 Reduce to lowest terms a2+3a+2 7* 11. 13. 6a;2+a;— 1 4x2+8x+3* og+2a+l a3+2o2+2a+l ' a^-4a+3 4a3-9a2-15a+18' 2a:3— a:^+2x— 3 2a;3+3a;2+4a;+3 " gg+gg— 3g— 3 g5irct*^a3+2g2— 3a— 5 ' /2. 4. « 8. 10. 12. 14. x^-\-5xy — 24i/2 x^-{-2xy-\-y^ — z* a;2 — y^—2yz — a* * 2-3j/-2y2 4_5y_62/2 * a;3 — x^—2x a;3-3a;2+4 * _3x2— 3x— 18_ 6a;6-12x*-18a;3" 3x3+4x2— 6a ;— 8 36a;3+27a;2_40a;-16' 2a:«— 4a;3— 2a;2— 12 a; 4a;*+2x3+6a:2— 4x~ 138. Addition and Subtraction. In adding or subtracting fractions we should be careful to note whether any of the given fractions can be reduced to lower terms. When the result is obtained we should examine it to see if it can be reduced. Ex. 1.— Simplify — ^ + 2y xy'^+y^ y The expression = + x^—xy^ y'{x+y) 2y x—y x{x—y){x+y) ^P~..~ x—y X ^x—y X """ x—y x{x—y^ ^ {x—y)*+2xy-y* ^ x(x—y) _ x* _ a; ~ x(x—y) ~ x—y' The form of the last fraction in the given expression should prompt the Dupil to examine whether it can be reduced. 192 ALGEBRA Ex. 2. — Simplify + x—2 x^—3x-\-2 a;2— 4a;+3 The expression x-2 ' {x-l){x-2) (a;-3)(x-l)' _ (a-l)(a;-3)+x-3-2(a:-2) {x-2){x-l){x-Z) _ g*— 5x+4 _ (x—4)(x~l) x—i (x-2){x-l){x-3) {x-2){x-l){x-3) {x-2){x-3)' Simplify BXERCISB 94 i.*_?_^__L. 2. — —. 3. ^+y »-y a+ft a— 6 ' z— t/ x-\-y ' x—y x-\-y a— 4 o— 7 , 2a2 2a a' a— 2 a— 5 o*— 6^ o+6 * a—a^ 1+a' yf 2x' 2x2 ^ ^_y 1 ^ x'-4y' x-2y x*— 2/2 x^+xt/ r ' x^— y2 2x— t/ * x*+2xy x ,<ft^ __i ^ .j^ ^ 3y I ^'+y' ^JC x*+9x+20 x2+12x4-35 lAll5 jj+y x-y'^ x^-y* ^ x2-3x-10 x2+2x-3 -TT; x-y 2x x'+x'y «• x2-8x+15 x2-3x+2' i^ y x^ x^y-y^' ^^ a'—ab+b^ _ a^+ab+b^ ^ ^ x+y izy _ y—x ^. - %^ a—b 0+6 V^* x—y x^—y^ x-\-y \\jf a-fe I a+6 _ o^+fc" Cj-' 3x^-8 _ 5x4-7 2 ^' 2(o+6) ^ 2(a-6) o^-ft^ ' Af^" x^-l x^+x-j-l """x-r ^' ^_+i£±^ + _L_. i«^ J___1_ + _L., 2a-36^4a«-962^2a+36 k^ x-1 x-2^x-3 «-.-d=.* * ^ x»-3x+2 x2-7x+10 x2-6x+5 (o4-6)(6+c)(c+a) _ a+fe _ 6+c _ c-fa a6c c a 6 FEACTIOHB aj a«— 6*+26c— c« cg+2ca+o^— & * ^. az-{b-c)^ b^-{c-a)' c^-ia-hY' •' (o4-c)2-62 (6+a)2-c2 "^ (6+c)2-o2 o* — 2a 3a 5a oa-a-2 6a-4 6o2+2a-4 25. and x—y x+y x-y x+y x^+y* :6;I_i 1 ?^ i^. ^ x-t/ a;+?/ x^+y^ x*+y* 1 1 2x 27. 3— a; 3+x Q+a;^ ^28 , « I « 2a=» 4a'6« • >• a-6 "^ a+6 "^ a^+fc^ "^ a*-b* 1 1 X X 4— 4x 4+4x 2+2«2 1+z* '"• ^"^"^ x^lk+6+ x^+lt+U - x^+ll+2l = l' <^^^y-^ 139. Special Types in Addition and Subtraction. We have already seen that 6— a= — (a— 6) and a—b= — {b—a), cr (a— 6)-i-(6— a)= — 1. When a— 6 and 6— a oecur in the factored denonainatoM Df different fractions, which are to be combined, it is not leeessary to include both of them in the L.C.D. a b Ex. 1. — Simplify r + ? • ^ -^ a—h b—a Here only one faetsr is required in the L.C.O. ftad we may uie either a—b or b — a. a we de<nde to vise a — b, then it is better to change the second 6 fraction into the form r • a—b „, a b a b a—b Then — i + r = » ; = r -= I. a_|, b—a a—b a—b a—b 194 ALGEBRA t:. o o ir 4 3 , X^S Ex.2.-S.mphfy--^-^j + j-^,. The denominator of the last fraction should be changed to a;*— 1, BO as to be the product of x— 1 and x-\- I. rru .43 x-3 The expression = - — — r z — ^ , '^ x—l x+l x*—l 4(a;+l)-3(a;-l)-(a;-3) 10 (x-lKx+l) (x-l)(x+l) Ex. 3.— Simplify 1^2 3 {x-l){x-2) ' {x-2){3-x) {x-S){l—x) Here there are only three factors x—l, x— 2, x — 3, required in the L.C.D. We therefore change the second and third fractions so that the given expression 1 9. a + (x-l)(x-2) (x-2)(x-3) ' (x-3)(x-l) Complete the simplification. 140. Cyclic Order. Suppose we wish to simplify 6+c , c-\-a a-i-b {a—b){a-c) ' {b-c){b-a) ' (c-a)(c— 6) The L.C.D. in this case will contain three factors and it might be written in dififerent forms as {a—b){b—c){c—a), {a—b){a—c){b — c), etc. The pupil is advised to write the factors in what is called cyclic order It we arrange the letters on the circumfer- ence of a circle, as in the diagram, and follow the direction of the arrows we see that a is followed by 6, b by c, and c by a. Thus, if we write a — b as the first factor, then changing a to 6, 6 to c, and c to a, we write the second factor b — c and the third c — a. If we write the L.C.D. as (a — 6)(6 — c)(c — a), we should change the fractions so that these factors appear in the denominators. FRACTIONS 195 'Rie given expression then _ b + c c+a a+b ~ ~ (ar-'b)(c -"7) ~~ {b-c){a-b) ~ (c-a){b-c) ' _ — (6+c)(b-c ) -(c + a)(c— g)— (a+6)(a— 6) ~ (a — b){b — c){c—a) — b* + c* — c* + a^ — a' + b* (a—b){b—c){c—a) = 0. „ o- ivc be , ca at> Ex. — Simplify j— + -j- —r- + {a-b){a-c) {b-c){b-a) ' {c^cjjic-b) Proceed as in the preceding example and you should get the result — 6*c+6c*— c*a+ca*— a*6+a6* (a=^(6-c)(c-a) This fraction is equal to uniiy, for the numerator is equal to the denominator. Prove that this is true. EXSRCisE gs Simplify : 1.* 1 1 a^+ax x'+ax „ x+3_x—3 r' x-2 2^' 5. a^—ab b^—ab _2 3 x2--3 X—l X+1 1— X2 /^ x—y x-\- + y y x+y y^—x^ a;+4 x— 4 16— x* ^2 3a+2x _ 3a- 2x 16x2_ */ 3a-2x 3a+2x'^ ix^-9a^ j3 ^^ 4 _8_ 3a:+7 j' x-1 1-x i+x^x2-r • > IM ALOEBRJi 14 -J^ i 'a-h){c-a) ' (6-a)fc-6) 16. , f^— + * (x-oXa-fi) (x-6)(6-o) 2 2 1 17. -^— , + .., » , + a 3-x 3+x l-16x l-3x l+3x ' 9x2-1 ■ 2 2 a;2-8x+15 x2-4x+3 Gx-x^-S 20 <^+^ 1 fe+c J f +a * (6-c)(c-a) "^ (c-a)(a-6) "^ (a-6)(6 -•! 21 1 , L I 1 • (a-b){a-c) ^ (b-cyh-a) ^ {c-a){c-b) ' n.^ 7)2 «2 /22. ?^ + - + ^ / (a-6)(a-c) (6-c)(6-a) ^ {c-a){c-b) ^ ' {x-y){x-z) (y-z){y-x) {z-x){z-y) ax— be bx—ca cx—ab ~r 7T — TTTT , + (a-6)(a-c) (6-c)(6-a) (c-a)(c-6) ' a* 62 g£ (/^^ • (a2-62)(a2-c2) "^ (62_c2)(62_a2) "^ (c^TT^sj/^lTj;^ / ■ (a-b){a-c)~^ {b-c){b-a)'^ {c-a){c-b)' 27 1 1 _ Q+36 463 ■ 0-6 2(b+a) 2{a^'+b^)^ b*-a'-' Va;-6^x+5/ Va;+3^x-3 29, — ^4--^ L (Check when. a- i.) »+4 a+3 a+2 a+1 ^ 30. _L-|-_L 1 L. (Checi£whena;=2.) «— d z+3 X— 1 x+l FRACTIONS 197 141. Multiplication and Division. The ordinary cases in multiplication and division of fractions have been treated in art. 74. Some special forms which appear are illustrated in the following examples. Ex. 1. — Multiply a -\ by a — - • Here the mixed expressions should be reduced to the fractional form before multiplying. _, , a*—ax-\-ax a*+ax—ax o* The product = ' x — a+z a* — x' Ex. 2.— Multiply ? + - + ! by J+--1. a a Multiply thia in the ordinary way, by multiplying each term of th« one by each term of the other. We should recognize that the first expression is the sum of t H — and 1 and the second is the difference. Th» produce - (I + 1)'- 1 = ?,■ + 2 + ^^: - 1 _ «-! + ■ + ^^:. Or, we might proceed as in Ex. 1, thus ; a*j^b* + ab a* + b*-ah ^ {a* + b*)*-a*b* _ a*+a*b*+b* ab ab o*6' ~ a*b* a* 6* This result is seen to be the same aa i-i + 1 H — j, and the answer 0' a' may be given in either form. Ex. 3.— Divide ^ + - by ^-- + -. y X y y ^ The dividend = ^^^^, the divisor = «^*-^+y* :. the quotient = "^ x , '^'^ , = ^±1^ • xy' x'—xy+y^ y Divide in the ordinary way and get the quotient - -f- 1. We must not make the error of tliinking that we can invert the divisor, or take the reciprocal of it, by inverting 198 ALGEBRA each term of it, and change the problem to one in multi- phcation, thus : The reciprocal of ,, is a-\-b, but the reciprocal of - + r is not a-\-b. a _, 1,1 a-\-b - .^ . , . ab ± or — ^ - = — r— and its reciprocal is =- • a b ab a-{-b- Multiply : ^^ x^—x—6 x^—2x —8 ' a:2+4x+4' a:^— 7x4-12" »M-2x-15 x^+lx- U ' x^+8x—33' x^-{-9x+20' ^^ a+b a—b EXERCISE 96 2. 4. a'^+b^ ab-b^ a a^—ab' a*—b*' b ' r*-b* a-b a' a2— 2a6+62' ^2^06* (Py + ^,!j ^ y *-«" a;2+l+-, x2 x^ I + -2 I/— a; a;+y' y^ + x* 2 - - 2 ~2 ■ 9. 6 + ^^, 1 Divide a+a;' 6a; ^ a^ + 2 + ^2, a2 - 2 + 11 a;^-lla ;+30 , o^—5x a;2_6x+9" "^ a;2-3a; ' 12, ^. o2+62_g24.2a6 , a-\-b-^c c2— a2— 62+2a6 6+c— o y b i+rby? + ^, ' y3 x' y X iplify: o*-4 0+1 0+2 a^—2a a^—a ' a*—a^ a" a a^-b^ ajrb a^-ab+b^ J3+P' a-b' o2+a6+62' o2._62 ^ a^-\-ab by x^—xy xy—y I M* .<9. ^-^->y^'-^' ^^^ x*—y* x—v x—y . gg— 4 . a''+2g •'^' o2+5o ■ o2— 25' FRACTIONS 199 XJ^ 2x2+x-l 2x^-5a;+3 . 2x^-7x+6 ^' a;2-4a;+3 ^ 6a;2+a;-2 * 3a;2-7a;-6* >. (— -^ + ^.)-(«+-)^- ^ (a+6)^-(c+t^)^ ^ (g-c)2-(rf-6)'^ ^' (a+c)2-(6+d)2 • (a_fe)2_(d_c)2- 23 a''-64 02+120-64 ^ a2-16a+64 ' a2+24o+128 0^-64 " a2+4a+16 ' 24 A ^ _ 26 \ ^ / 2a _ 26-a \ • V0-6A 3a-26/ ■ U+26 26-3a/ 142. Complex Fractions. A complex fraction is one which contains fractional forms in either the numerator or denominator or both. Thus, - is a complex fraction and is, of course, onky another way of writing r -f- 3 • It is simplified in the usual way by changing it into a d , . , , ad ,- X - which equals r- • be ^ be A complex fraction may sometimes be easily simplified by multiplying both terms by the same quantity. Thus, — r — = —TT — on multiplying each term by 4. ii ^•»- a+2b a a + b "^ b _ 6(a + 26) + a(a + 6) _ a*+2ab + 2b* _ a+2b a_ (a+2b){a-\-b) — ab a*+2ab + 2b* b ~ a+b Here both terms were multiplied by 6(0+6). 500 ALOE BR A If th« L.C.D. is not the same for both terms of the fraction, it is usually better to simpKfy tko terms separately. a;-|-l x—l Ex. — Simplify The numerator = The denominator = .•. the fraction = X-f2 X—2 z—2 x+3' x-{-2~ z^ {x+l){x-2)-{x-\){x+2) — 2x (x+2){x-2) (x+2){x-2) (a;-2)(z-3)-(x+2)(x+3) -10a; (x+2)(x-3) (a;+2){a;-3) -2x (a;+2)(a;-3) x-S (x+2)(x-2) -lOx 6(«-2) Simplify : 60 66 BXBROISBl 97 1.* 6. 10. 12 12c 2. 1 1+* 1- l+x o — 2 — a^—5a a— 3 a + a— 3 2 + xy + — m xhj[^ — 1 t a \ _a 6_' t-fc c-H» 60 12c 6. 3. 1 g— ft a+6 4. 0+6 0—6 1 ] a-j-6 a— 6 a; + y — 11 a; y x—y x^+y* 7. a;4-y x + y 2zy_ x-\-y g (a;+3y)g-(a:-3y)' ■ (Zx+y)^-{Zx-y)* IV 13^ + 6- "o2- 03 -oi+fea a + b- fe8 0*- 3 -o6+6» 2x4-3 3 I- X4^ FRACTIONS 20% a b 1*. I . 16 " — I — — — I — — a — a+b a+b a^ ab^ b^ 16. Find the value of -^^ when x= -^ , y= ^ . x+y a—b a-\-b 17. Find the value of ^~^^+? when a - ^11^, 6 = fc^' l+2a+6 x+2/ (a:+2/)2 EXERCISE 98 (Review of Chapter XV) Simplify : a ^ a a 2a* a b c o _« y (a?-!/)* - 2 y ac_ /^' a;+3y Sy-a: a:»-9i/»' * l_lj_!/ ^_x X y X y /K ^_ ^ _i_ 4. ^_ 4- ^ fi ^_ _ 1 L- ^ 1 7 /i . ("-fe)' ) ^ / 1 I fe' + Q* \ J, (l-a;)3 + (l+x)=' *• r"^ 4a6 J • V 2o& /■ (l + a;)»-(l-a;)»' „ / 26c \(, , 26c V, 6c-c»\ 6* _1 . 1 1 x+1 a;»+3x+2"^a;3+6a;»+lla;+6' a*—ab ab + b* 2a»6 a'-b» ' o» + 68 ' a*+o»6«+6« o«— 6» a* + 6* a+6 t/^ • a«+2a6 + 6» ^ o»^^» ^ o*+o*6»+6* ' v'o a+x a—x 2x' i±_+ - - + zz +oa;+a;* a*— aa;+a;* a*+a*a;*+x* yi4. (l-a«) - |(l-a)« - (a>-l) - (a + ^)}. 6+c— o c+a— 6 0+6— c W«^* (o-6)(o-c) ■*■ (6-c)(6-o) ■'" (c-o){c-6) * 202 ALGEBRA '16. ^* + y' 1 '' . ix-y)ix-z) {y-z){y-x) ^ {z-x){z-y} 47 ^+6 3x-9 ISx-lO '• 6z» + 6x-6"^ 2a;»-3x-9 9x»-12x+4" 18. Express the product of 8-6x + a;« ^ ) 1 i and l+x (a;_i)(2-a;) (x-2)(4-x) (a;-4)(l-T) as a fraction in its lowest terms. 19. How can you show mentally that 3 is the sum of XXX a , b I ' c x+a x+b x+c ^^ x+a~^x+6 x+o 2x_ _^ y*_ +y x—y x*—y* "^ x + y ' x»-t/* Divide -=P- -\ ^ ^^ — - by - ^^^ f- - x+w x—y x*—y* •' x + y x ^. ., a*-(6 + c)* 6'-(c-a)« , (a-fe-c)» 4^' f' '22. Divide ^ + '-^ + ^-:::i-i by 2-fl+UlY / a b c \a b cl 0/f, Show that ,/'^+^^^^-^^ = x(l-y«) + y(l-x') ^ ^ (l-X3/)»-(x+t/)» (l_a;«)(l-2/»)-4x2/ o. Tf 2,2 2,2 24. lfa = ;r — 7. o=r , c=i5 — 5. o=s , prove a = x. 2 — 2 — c 2— d 2— X ^ 25. Find the product of l+a:' J , 1 y-^, and 1+X+ „ 10 » " '^^' 26. Subtract . , -^ from t- and determine which fraction is the o-\- 10 greater if a is greater than b and if both a and 6 are positive. y Add (^+°^)(l+°g ) (l+bc)(l+&a) (l + cq)(l+c6) (a-fe)(a-c) ' (6-c)(6-a) ' (c-a){c-6) 28. Show that 1 =• has the same value when x = a-\-b as it x—a x—b has when x= j • a+b 29. Prove that the product of any two quantities is equal to their Bum divided by the sum of their reciprocals. *♦,><'/ FRACTIONS 203 h — c c — a a—b ^, ^ 30. If X — , y = — ; — , 2= , prove that xyz + x + y-\-z = 0. 31. If a=^^^, 6 = ^-^, c=^^=^, show that x+y y+z z+x (l_a)(l_6)(l_c) = (l+a)(l+6)(l+c). 32. If a and b are positive, which is the greater a+3 b a + 26 a+26 ^"^ a + 6 J ^^ a+3b _ 4b» -6 2(a+6) 2(a* + 6*) o*-6*' »y33. Simplify - x~y y—z z—x 2*— (a;— 2/)*' a;*— (y— z)*' i/* — (z— a;)* 4 3 12 35. Simplify r ^ -| , ^. ^ ^ x—t X— 3 X— 1 X— 2 36 When x= , find the value of a+c x—2a _ x+2a iac x+2c X— 2c X*— 4c* r , «— a; y— x ^ 37. Simplify J _i±^-^;-^ I, -,(^-?\ j ^ x(y-x) ^ 2/(y-a;) \a; «/> CHAPTER XVI FRACTIONAL EQUATIONS 143. If an equation involves fractions, the fractions may be removed by multiplying every term by the same quantity. In Chapter VI. simple examples of fractional equations were given. The case in which two fractions are equal deserves special attention. d c Thus, if r = 3 and each side is multipUed by bd we have a - X bd = ^ X bd. a ad=bc. 144. Cross Multiplication. It is thus seen that when two fractions are equal, we can remove the fractions by multiplying the numerator of each fraction by the denominator of the other and equating the results. This operation is sometimes" called cross multiplication. Ex. 1.— Solve z—5 x-{-3 x—1 a;+9 Cross multiply, (a;-5)(a;+9) = (x4-3)(x— 7), .-. a;» + 4aj-45 = x»-4x-21, 8a;=24, x=3. Verify by substitution. This method is applicable only when a single fractioD appears on each side of the equation. FRACTIONAL EQUATIONS 206 r. « o , 4a:+17 ^ 3a:- 10 Ex. 2.— Solve — —r- = 7 — . x4-3 X— 4 Simplify the right-hand member and we have 4a;+17 _ 4a;— 18 a;-f3 ~ a;— 4 ' Now cro8S multiply, complete the solution and verify. SXBRCISE] 99 Solve and verify : ^ 3a;+l _ a;+12 2 3a:-5 3a;-l 2 3~' * 5x-Z~ 5x-\' „ a; a;— 1 „ . x— 2 a;— 6 ^ io. = a; -- 9. 4. — = 0. r 3 11 a;-3 x-5 L x4-2_a;— 2 a;— 1 ^ x+1 _ x— 3 _ a;+30 ■ ~l 2 1~' ~2 3 13" ■ _ 7a;-3 , , 2x+5 „ a;2_^7x-6 x+1 / 2 3 x2+5x-10 x-1 9. ^ = 2-^. 10. J_ + l = ^. X— 1 X— 3 X— 1 X x-(-l 11. J_ + J_^^. 12. ^i:i_2^ = o. x+2 X— 2 x-3 3x— 2 6x— 1 13. 2x+38 _ 6x+8 ^ ^^ y-S _ y-l2 49. x+12 2x+l 2/2-87/+15 2/2-122/+30 ^^ 2x4-7 8x+19 5x+ll ^^ X— 1 , x-5 „ 10. = . lb. = ^. 3 12 7x+9 X— 2 x-3 ._ 6-8x , 3 „ ^„ 2x+7 , 3x-5 5x+9 17. = 8. /IS. ■ — — . r 3— X 1— X / x+1 x+2 x+3 4x3+4x2+8x+l _ 2xg+2x+l 2x2+2x+3 x+1 X— 3 X— 5 /21 ^-^-^ J- _^ _ ^ = 2x-3 * 1-6 "^1-25 1-8 9 "^ "■ 206 ALOEBRA oo c- 1 3a;«— 4a;+5 4a;«— 5a;+7 . -, ^ j • u • . 22. bolve ■ — = — by farst reducing each iraction 3x— 4 4x— 5 to a mixed expression. 23. Find three consecutive numbers so that the sum of ^ of the first, I of the second anu } of the third may be 30. 24. Divide 300 into two parts so that if one be divided by 5 and the other by 7, the difference of the quotients will be 18. Give two answers. 25. How much water must be added to 100 lb. of a 4% solution of salt to make a 3% solution ? 26. A pupil was told to add 3 to a number and to divide the result by 5. Instead of doing so he subtracted 3 and multiplied by 5 and obtained the correct answer. What was the number ? 27. A man bought 180 lb. of tea and 560 lb. of coffee, the coffee costing 1^ as much as the tea per lb. He sold the tea at a loss of 25% and the coffee at a gain of 50%, and gained $62-60 on the whole. What did the tea cost per lb. ? 28. I sold some butter at 25c. a lb. If I had received 5c. more for 1 lb. less, I would have received 2c. more per lb. How many lb. did I sell ? 29. If I walk to the station at the rate of 11 yards in 5 seconds I have 7 minutes to spare ; if I walk at the rate of 10 yards in 6 seconds I am 3 minutes late. How far is it to the station ? 145. Fractions with similar Denominators. ^ .. CI , ^+6 2x-18 2z-\-3 16 , 3a;+4 Here we might multiply each term by the L.C.D., which is 132. It will be found simpler, however, to remove all the fractions but the first, to the same side of the equation, as they are easily reduced to a common denominator. g+6 _ 2a;— 18 2a:+3 16 3x+4 n " 3 4 "^ 3 + 12 ■ x+6 ^ 4(2a;- 18j-,3(2a;+3)-l:64+3a;-|-4 • U ~ 12 Now simplify, cross multiply and complete the solution. The ettrrect answer is a; =6. FKACTWNAL EQUATION if 207 This problem shows that the denominators of certain fractions are such that these fractions can be conveniently combined when they are grouped on one side of the equation. T. o o , 2a:+3 4a:+5 , 5a;+4 EX. 2.-Solve __=.__ + __^. Since 4x+4 = 4(x+l), it is seen that it is simpler to combine th^ second fraction with the first than with the third. 2x+3 4a;+5 _ 5a:+4 x+l ~ 4{x+\) ~ 5x-{- 1 * Subtract the first two fractions, complete the solution and verity the result. T^ ^ o . x—l x—2 x—4: , x—5 Ex. 3.— Solve ^ o K + « = 0- X — 2 a:— 3 x—o x—o Here it is too laborious to multiply all the fractions by the L.C.D. It will be found easier to change the equation so as to have two fractions on each side, then simplify each side and cross multiply. Solve by transposing the last two fractions, also by transposing the second and fourth, and compare the results. Ex. 4.-Solve -1^ + -1_ = ^-1-^ + -1-g . AA^- u -J 2a;+7 2x+l ,. Addmg on each side, ^Tpf^+To = ^^+T^c+12 ' ^^> 2x+7 2x+7 _^j^ x*+7x+10 x^+lx+V2 •'• ^^"'^ '^ (x^+lx+lO ~ x^+lx+u) = ^' x=-S^ or x^ + lx+lQ = x' + 7x+\2. Since the equation x* + 7a;+ 10 = x^+7a;+ 12 is impossible, the only root of the given equation is x= —3^. (Verify this root.) If in line (1) we divide each side of the equation by 2x-\-'i an impossible equation will result. It is not allowable to divide both sides of an equation by a common factor unless we know that the factor is not zero. Here 2x-\-l might be equal 208 aW£:BRA to zero, and, in fact, would be if x= — 3^. If x=—Z\ the equation in line (1) is satisfied as each side becomes zero. Solve the equation by writing it in the form ji L--J L X-+5 a;+4~z+3 x-\-2' EZBRCISB 100 Solve and verify : 2x+l _ 6x— 1 _ 3a:— 2 ^ 2a:+3 _ x—\ _ x-\-2 5 ~T5 6a;+3' ' 4 6x^ ~ 2 a;+3 3a:+5 _ 2a:4-l 6x4- 1 3x-l 2x-l * "t" 6x+2 ~ 14 ■ 4 2 3a;-2 R * I 3 29 , ^ =ft. @ x-8 2a;-16 24 3x-24 _5 5a;— 5 _ 6x4-7 12 "*" 12x+8 ~ 9x+6 ■ 13x— 10 4x+9 _ 7x-14 _ 23x-88 36 ~T8 f2~ ~ 17X-66 ' 3x-4 1 6x-5 6x-9 12 8x^12 5X-17 , 2x-ll 23 3x-7 18-4x 14 42 21 1 5x-7 4x-3 10 lOx-5 4x-2 Jl l___l 1^ X— 1 X— 2 X— 3 X— 4 1111 X— 10. X— 5 X— 7 X— 2 r^. -i^+ ' ' 14. 3x-fl2 6x+24 2x4-10 x f 6 3 5__ _8 1_ —5 X— 7 7— X 5— X X— 8 , X— 4 X— 5 , X— 7 ,, , . 15. 1 = 1 (transpose terms) ?;— 10 X— 6 X— 7 X— 9 FRACTIONAL EQUATIONS 209 ,^ 2x-21 , x-1 x-12 , 2x-n lb. ■ = • x-14 x-8 x-13 z-9 2x— 27 ' 17. Solve the preceding example by changing into 2 + x-14 ' x-14 and making similar changes in the other fractions. ._ 4x-17 , lOx-13 8x-30 , 5x-4 19. X— 4 2x— 3 2x— 7 X— 1 5X-64 2x-ll 4X-55 x-6 X— 13 X— 6 X— 14 X— 7 x+l^x-2^x-l 21. Solve \ 1 — - — 3. b-\-c c+a a+b 22. If a^—b^—a—b, does it follow that a must be equal to b) What is the alternative conclusion ? 146. Literal Equations with one Unknown. Equations often occur in which the known quantities are represented bj" letters instead of numbers. These are called literal equations. The same methods are used in solving them as were used in solving equations with numerical coefficients. Ex. 1. — Solve az=bx-\-c. ax~bx = c, .*. x(d—b) = c, _ c a — b Solve 8a; = 3>c+20 8x-3x = 20, .-. 5x=20, .-. x=^ = 4. Here the letters a, b, c represent some known numbers whose values, however, are not stated, while x represents the unknown whose value is to be found in terms of a, 6 and c. Usually the earlier letters of the alphabet are used to represent known quantities, and the later ones x, y, z to represent unknown ones. Compare the two solutions given. They are practically identical. When we work with numerical coefficients the result can usually be expressed in a simpler form. P 210 ALGEBRA Note. — The pupil must not make the mistake of giving x= as a sohition of ax = bx-\-c. This statement is true, but it is not a solution, since it does not give the value of the umknown in terms of known quantities only. Ex. 2.— Solve a{x—2)-h = a—2x. Removing brackets, ax—2a — b = a—2x. Transposing, a^-[-2x = a+2a + 6, .-. a;(a + 2) = 3a-t-6, _3a + 6 "^"0+2' The result should be verified by substitution, but this will frequently be found more troublesome than the solution. When it is not verified in the usual way, tue pupil should review his work to ensure accuracy. T^ « r. 1 ^—^ x—a Ex. 3.— Solve a—x h—x Cross multiply, bx—b^ — x^-\-bx = ax—a*—x^-\-ax, :. 2bx-2ax = b'-a', _ ^'— °' _ ^+a ^~2{b-a) 2" Verify by substitution. EXERCISE 101 Solve for x, verify 1-12: 1. mx-\-a=b. 2. ax=bx-\-2. 3. a-\-- = c. h x-{-a _7 x—a_a _ x—c_a—b x—a 3 " x-\-b b ' x-{-c a-\-b a_ 6 _ x—a_x—b . x—ax—h_. X x—a+b. ' x—b x—a ' 26 2a * .. ax c b dx ** I \ i Ti ■L\ c\ 10. — = . 11. o(a;— o)4-6(a;— o)=0. d a c l_ 1_ 1 a X X b 12. 1 _ 1 = 1 - 1 13.* (o+a;)(6+a;)=(c+a;)(d4-a;)- FRACTIONAL EQUATIONS 211 14. {ax-b)(bx+a)=a(bx^-a). 15. "J^^ - ^-^"^ =x. 16. a b _ a—b x—a z—b x—c 17. x{x—a)+x{x—b)=2{x—a){x—b). 18. {x-a){x-b) = {x-a-b)^. 19. J '-^^ '-. X — a X — 2a x — 3a x — 4a 20 {x—a)(x—b)-(x+a){z+b) = {a+b)^. .21. {a+x){b+x)-a(b+c)^'^ + x^ b 22. (a4-z)(6-x)+x2=fc(a+x)- — . a 23. a'^x^b^+abx=a^—bH. 24. The excess of a number over a is three times its excess over b. Find the number. 25. Divide the number a into two parts so that one part may contain b as often as the other will contain c. 26. Divide a into two parts so that m times the greater may exceed n times the less by b. 27. A rectangle is a feet longer and b feet narrower than a square of the same area. Find the side of the square. 28. If a number be divided by a, the sum of the divisor, quotient and one-third of the number is 6. Find the number. 29. A man sells a acres more than the mth part of his farm and has b acres more than the nth part left. How many acres were in the farm ? 30. Solve {a—x){b—x)={c—x){d—x). Check by putting a=l, 6=6, c=2, d=Z. 31. If 5 = - (a+i), solve for w ; for a ; fori. 32 . If 5 = , solve for a ; for I ; for r. r— 1 33. If 8=at-\-^gt^, solve for a ; for g. V 2 212 ALGEBRA 147. Literal Equations with two Unknowns. Every simple equation in x and y may be reduced to the form ax-\-by=c, where a, b, and c represent known quantities. If two equations in x and y with literal coefficients be given, the equations may be solved by the same methods as were used with equations with numerical coefficients. Ex. 1. — Solve ax-j-by—c, ax—by=d. Adding, 2ax=c-\-d, .". x = -x c—d Subtracting, 2by=c-d, ,". y 26 Verify in the usual way. Ex. 2. — Solve ax-\-by=c, (1) 'mx->rny=k. (2) Multiply (1) by n, nax-{-'nby=cn. Multiply (2) by b, binx+bny=kb. Subtracting, x(na — 6m) = cn — ^6, _ cn — kb na—btn We might substitute this value of x in either of the given equations to find y, but it is simpler to solve for y in the same way as we did for X. Eliminate x from the two equations and find y = , or = — . bm—an an—bm Ex. 3. — Solve a^x-^b^y=c^, a^x+h^y=c.^. Here the symbols a,, a.,, etc., are used to represent known quan- tities. They are read " a one, a two, b one, etc." There is no relation in value between a^ and a.,, nor a, and fej. The notation is used to obviate the necessity of employing many different letter forms. Solve these equations as in the preceding example and obtain ^ ^ 62Cj-6iC^ ^ ggCi-aiCg 016,-026,' ftjOa-ft^Oi FRACTIONAL EQUATIONS 213 EXERCISE 102 Solve for .r and y, verify 1-12: 1. mx-\-ny^=a, 2. lx-\~my=m, 3. px-\-qy=r, mx—ny=b. ' mx-\-ly=l. x-|-j/=0. ^'■■ 4. ax-\-hy=a^-{-h'^, 5. ax-}-by=2ab, yji. ax^hy=2, x-\-y=a-\-b. bx—ay=b^—a^. ahc—b^y=a—b. J. ax—by=2a^+3b% 8. ax—by=2ab, bx+ay=—ab. 2bx+2ay=3b^-a^. ^ 9. a^x+bh/^a^—ab+b^, g^ ax — by=a — b. 10. "^ + 1 = 3, ^1. - + f = 2, 12. a^x-b^y=a-'+b^ n ^ n. h a b ^^ a b a b X y , ax—by=a^—b^. r = 2. 13.* a2x+62y=a3-63, 14. a^x+biy=Ci, 16. _ + _ = _ ^, 3a 46 ^, X y ' 16. (a+6)a;-(a-6)l/=^a2+6^ 17. ^ + ^=c„ x—y=a—b X y X y 18. a: + ^=^ + 2/ = 6. 19. l:^ + ^==22. a — • a-\-b X y •5a '356 _rt_ a; 2/ 20. If ax-\-by=c and x— 2/=l, prove that x{c—a)=y{b-\-c). 21. If y=ax-\-b and x=py—q, prove that y{q—bp)=x{aq — 6). 22. If bx-\-ay-\-cz=ac-{-bc, ax-{-by=ac, cy-\-dz=ad, solve for x, y and z. 23. What is the value of m, in terms of a and b, if the following equations are consistent ax+362/=a2+362, 3x+i/=3a+6, 4tx~Zy=m ? «14 ALGEBRA BXE3RCISE;(39i§i^ mevlew of Chapter XVI) Solve and vnrify 1-24 : A 5,7-.,-„i.-.)=,(«-8). A- ifEi-:±f;=i- -xy_^xy_^ 2x+] _ 9a;- 8 _ x-U ^ 2"^ 3 »' 3"^ 2 «■ ^- 7 11 ~ 2 ' /2x-3 3x-2 _ 5x»-29a;-4 m x+1 _ 3 _ x 5-x • x-4 "^ x-8 " x*-12x+32' ^' ~^ x~3 6~ * 5+3x _ 4x-7 _ 16x-27 _ x+3 - 2x-3 3x-7 _ 2 3 ~ 21 5 V"- 2x+l + 3x+5~ • 1^. a(x— a) — 6(x— 6) = (o + 6;(x— o— 6). 12. ax+6 = 6x+a. ^, 8x + 5 9x-3 4x-3 ^, 4-x 2 + 4x ^ ,/^- -To"- 7^+2= -5-- /*• "13 -7— = ^-^- 16. - + f=l. f-f,= 5. >6. «^J = £^. a 3a 40 -^ ax—b ex— a 17 A4-l-7l + -i--3 ^18 ^ - 1 I ^^ . tf X b 19. ax + 6i/ = 2o6, aw-6x=o»-6». 20. — r , =1 -r — y^ ' ^ ^ x + 6 — a x+6 — c 21 ^±^_^iJ_^+^_^±^. 1^22. ^ I ^ ^ y x+3 x + 4 xH-6 x + 7 • x+10 ' x-10 x-2 oq. ^Hi _ ^iJ _ ?Z:5 x-7 t/ x+1 _ 1 8x-3 V" x-2 x-4 x-6'^x-8 "• ^^' 3x-4 6'^15x-20 (i^5.* (3a-x)(o-6.) + 2ox=^46(a+x). '\x X 2x A/26. \-2b(a-c) + r = c(a + b)-\ ^ a c 27. 2x + ^=^-±-3+7.?^-«^ = 24i-4j,. 28. 1+2^-3x^3 7x-3y_^^^^ 2». - + -= -7, 3x-2y= -10a:y. FRACTIONAL EQUATIONS 216 30. ^ + ^ = 2lxy, 5y-2x = 24x7/. 31. What value of y will make x+5 , y—l , , x—l y+11 . -^+V equal to -^ + ?^? 32. Two sums of money are together equal to $1000, and 5J% of the larger exceeds 6J% of the smaller by 16 cents. Find the sums. 33. Find a fraction such that if 4 be added to its numerator it becomes equal to J, but if 4 be added to its denominator it becomes f . 34. If ax + b = cx + d, give the argument which leads to the con- clusion that X = , indicating at what point it is assumed that a a — c and c are vmequal. 35. Take any two proper fractions whose sum is unity. Add 'onity to the difference between their squares. Show that the result is always twice the greater fraction. 36. A man has §30,000 invested, part at 4|% and the rest at 5J%. He receives $65 per annum more income from the former than from the latter. How much is invested at each rate ? 37. The sum of three numbers a, b, c is 3036; a is the same multiple of 7 that b is of 4, and also the same multiple of 5 that c is of 2. Find the numbers. 38. If 5 = - (2a + nd — d), solve for a ; for d. 39. If ax — by = a*-{-b*, x — y = 2b and a;* + 2/* = c, find c in terms of a and b. 40. A man can walk 2^ miles an hour up hill and 3J nailes per hour Jown hill. He walks 56 miles in 20 hours on a road no part of which is level. How much of it is up hill ? 41. A farm cost 3| times as much as a house. By selling the farm at 7|% gain and the house at 10% loss, $2754 was received. Find the cost of each. 42. In 10 years the total population of a city increased 11%. The foreign population, which was originally ^j of the total, decreased by 1160 and the native population increased by 12%. Find the total population at the end of the period. CHAPTER XVn EXTRACTION OF ROOTS 148. Square Roots by Inspection. In art. 65 we have seen that the square root of any trinomial, which is a perfect square, may be written down by inspection. We have also seen that every quantity has two square roots differing only in sign. Thus, the square root of a^-]-2ab-\~b^ is ±{a-\-b), and of a^—2ab-\-b^ is ±{a—b). ± {a-'rb)=a-\-b or —a—b; ±(a—b)=a—b or b—a. If we had written a^—2ab-{-b^ in its equivalent form b^—2ab-\-a^, it is seen that b—a is a root. It is usual, however, to give only the square root which has its first term positive, and we say that the square root of a^-\-2ab-\-b~ is a-\-b and of a^—2ab-{-b^ is a— 6 or b—a. EXERCISE 104 (Oral) State the square of: 1. —abc. 2. x+l. 3. -x-l. 4. 2a-36. 5. a^+l. 6. x^-x. 7. a+b+c. 8. a+b-l. 9. 2a+b—c State the square root of : 10. 16x22/2. 11. la^b*. 12. x^+2ax+a\ 13. a2— 2a+l. 14. 4a2-12a6 + 962. 15. 9x^—30xy-{-25y' 16. x2-f-x+J. 17. a*+2a^+a^. 18. 16x*-48x*+36. 19. x^+y^+z'' + 2xy+2xz + 2yz. 20. o2-f-624-c2- -2ab- -2ac+2bc. 21. ia^+9b^+l + \2ah— 4a— 66. EXTRACTION OF ROOTS WH 149. Formal Method of Finding Square Root. When the square root of an expression of more than three terms ia required, it is not always possible to write down the square root by inspection. Thus, to find the square root of 9x4— 12a;34- 10x2— 4a;+l. Here we could say that the first term in the square root is 3x2, and that the last term is either +1 or —1, but it is evident that there must be another term as well. Let us again examine the square of the binomial a -[-6, which is a2-(-2a6+62 The first term of the square root is a, which is the square root of a2. The second term of the square root, h, may be obtained in two different ways, either from the last term, h^, or from the middle term, 2ab. Let us now see how we could obtain the second term in the square root from the middle term 2ab. This term is twice the product of a which is already found, and of the last term of the square root which is still to be found. If twice the product of a and the last term is 2a6, then we can find the last term of the root by dividing 2ah by 2a, which gives b. The quantity 2a which we use to find the second term in the square root is called the trial divisor. Since a'^-\-2ah-\-'b'^=a^-\-h{2aArh), we see that the complete divisor is 2a-\-h, that is, the trial divisor with the second term in the square root added to it. The steps in the process are : a*-\-2ab-\-b* \o.-{-h (1) The square root of a* is a. The a* square of a is subtracted from the expression leaving 2a6 + 6*. 2a + 6 | +2a6 + 6» (2) The trial divisor for obtaining the 2ab + b* second term in the square root is 2a. When 2a is divided into 2ab the quotient is 6, the second term in the root. (3) The complete divisor is 2a +6, and when this is multiplied by b 21 S ALGEBRA and the product subtracted from 2ab-{-b^ there is no remainder. Tlie square root is thnn 17 + 6. It might be thought that step (3) is unnecessarj-, as the root has already been found in (1) and (2). It is unnecessary if we take for granted that the expression is a perfect square. If you attempt to find the square root of a* + 2a6 + 46* and do not go beyond steps (1) and (2), you would get the result a-\-b, aa before. This, however, is not the correct result. Why ? We can now extend the method to find the square root of a quantity of more than three terms. 9x*- 12x'+ 10x«-4x+ 1 | 3x»-2a;+ l 9x* 6z^-2x \ -12x3+I0x2-4x+l -12x»+ 4x» 6x»— 4x+I |6x'-4x+l 6x»-4x+l After finding the first, two terms in the root, as in the previous example, the 3x^ — 2x is treated as a single quantity and tlie second trial divisor is twice 3x^- 2x or 6x^ — 4x. The square root is 3x'— 2x+ 1. 150. Verifying Square Root. We might verify the result in the preceding example by writing down the square ol 3:c-— 2x-4-l. Verify in this way. A simple method of checking is to substitute a particular number for x. When x=l, 9x<- 12r»+ 10x»- 4x+ 1 = 9- 12+ 10-4+ 1 = 4, ^nd 3x2- 2x+ 1 = 3-2+1 =2. Since the square root of 4 is 2, we presume the work is correct. EXERCISE 106 Find the square root, by the formal method, and verify the results 1. x2+12x+36. ?. 9a2-6a+l. 3. 9xH24x!/+16;/2. 4. 25x'^—\0zy+t/-. 5. \-\Sab + Hla'b'. 6. 49a*-28a^b^+ib*. 7. aH2a^-3a*— 4a4-l 8. 4x*+4x'+5a;«+2a;+l. EXTRACTION OF ROOTS 21© 9. z«-6a;3+17a;2-24x+16. 10. 9a*-12a3+34a2— fiOa+25 11. a*-4a36+6o262_4Q/,3^ft4^ ^12. a^-^a'^+^a^^. 13. 9a* + 12a36+34a262_(_20a63+256*. 14. x«-4x5+6x3+8x2+4x+l. 15. a;*-2x3+2x2-x+i. a* 4a3 2a^_4a 6*^ fc" ^ fe2 ft ^ , 17. o2_4a6+6ac+462— 126c+9c2. 18.* Simplify a(a+l)(a+2)(a+3)4-l, and find its square root. 19. By extracting the square root of x*+4x'+6x^+3x+7, find a value of X which will make it a perfect square. (Verify by substitution.) 20. If the square root of x*— 8x^+30x2— 56x+49 be x^+mx+T >vhat is the value of m ? 21. Using factors, find the square root of (x2+3x+2)(x2+5x+6)(x2+4x+3). 22. Find the first three terms in the square root of 1 — 2x— Sx^ and of 4-12x. 23. When x— 10, the number 44,944 may be written 4x«+4x3+9a:^+4x+4. Find the square root of the latter and thus deduce the square root of 44,944. 151. In algebra, an expression of which the square root is required is usually a perfect square. When such is the case the formal method may be greatly abbreviated. Ex. 1. — Find the square root of -r4-4a;3+10a;2— 12x+9. The first term is x' and the last is +3 or —3. The trial divisor for obtaining the second term of the root is 2x', therefore the second term is — 4x*-i-2x' or — 2x. .'. the square root is x* — 2x + 3 or x* — 2x— 3. 22t ALGEBRA If -(v* square ar* — 2x + 3, the term containing x will be twice the product of — 2x and 3 or - 12x. If we square x* — 2x— 3, the term containing x will be +12x. We thus see that if the expression is a perfect square, the square root is X* — 2x+3. Check this by putting x= 1. Ex. 2. — Find the square root of 4x4+20x3+ 13x2-30a:+9. What is the first term in the square root ? What is the trial divisor 7 What is the second term in the root ? What may the last term be ? What is the square root ? • (Verify your answer.) Ex. 3. — Find the square root of -3a3+-V-+a*-5a+fla2. Write the erpression in descending powers of a. The first term in the root is o*. The trial divisor is 2a*, therefore the second term is — 3a'-^2a* or —3a. .". the root is a*— ^a-fj ora* — 5a— §. Which is it ? (Verify by squaring.) Ex. 4. — Find the square root of 4 Here the terms are already arranged in descending powers of o, the term +9 coming between a and -. The first term in the root is 2a, the second is — 4a -r 4a or — 1, and the 2 la^t 13 +-. ~a Complete and verify. It will be recognized that it is only in the most complicated cases that it is necessary to use the formal method in full. It is advisable to use the contracted method whenever possible. EXTRACTION OF ROOTS 221 BXBRCISB 106 Find the square root, using any mt^thod you prefer. Verify the results. 1. x*+2x^-^x^-2x+l. 2. x*-4kx^+6x^-4x+l. 3. o*— 6084.502+ 12a+4. 4. x*+8x^+12x^—lQx+^. 5. 9a*-6a3+13a2-4a+4. 6. x* + 6xhj+7x^y^—6xy^+y* 7. 4a;*+20a;3_3x2-70x+49. x^' l-l0z+21x^-l0x^+x\ 9. 67a;2+49+9a;*-70z-30x3. 10. ai2_8a9+18a«-8a34-l. 11. a:«+2x3-a;+]. 12. x^-2x^+?,x^-lx+^^. 13. a«-6a2 + ll --, + 1 . a^ a* 15. - - 3x3 ^ i^a;2_^a;^i. 4 17. 54- 1 !_2x+x2. X^ X 19.* (x+i/)«-4(x+2/)3 + 6(x+2/)2-4(x+2/) + l. 20. x2(x-5a)(x— a)+a2(3x-a)2-3a2x2. 21. (a-6)2Ka-6)'-2(a2+62)f + 2(a*+6*). 22. (a4-6)*-2(a2462)(o4fe)242(a*+6*)- 23. (.. + J.) + 4(.'+i^) + 6. 24. If a;*+6x'+ 7x^^4-0x41 is a perfect square, what is the 7aiue of o ? 25. If the sum of the squares of any two consecutive integers be added to the square of their product, prove that the result will be a square. 26. If 4cX*+l2x^y+kx'^y^-{-6xy^-\-y* is a perfect square, find k. 27 . If m = X and n=^ y — -, show fjat X y mn4V(m2+4)(n24-4)=2x2/4- — • xy 28. Find the square root of 4x*48x348x244x41. Check when a:=10. 14. 4x* 4x3 3^2 2x y* y^ y^ y 16. 9o* 4a3 74a2 4o 25 "^ 5 "^ 45 "^ 3 ^ 18. y^ y X x^ 222 ALGEBRA 152. Cube of a Monomial. When three equal factors are multiphefl together, the product is called the cube of each of the factors. Thus, the cube of 2a or (2a)3=2a . 2a . 2a = 8a', the cube of a* or (a*)' =a* . a* . a* = a^, and the cube of Za^ or (Sa')' = ^^a^ . Sa' . 3a»= 27a*. The cube of a monomial is found by writing down the cube of each factor of it. Thus, the cube of 5ab*x is 125a'6'x*. 153. Cube of a Binomial. Find the c« + 2a6 + 6» cube of a-\-b by multiplying its square by ° "^^ a-\-b. Find also the cube of a—b. a^ + 2a^b+ ab* {a+bY=(i^+'^a''h + Zah-'-\-b\ + a^by,ab^+b^ Note that in each case the cube contains four terms, in descending powers of a and ascending powers of b, and the numerical coefficients are 1, 3, 3, 1. The cube of a—b is the same as the cube of a-\-b, except that the signs are alternately plus and minus. From the forms of these two cubes, the cubes of other expressions may be written down. Ex. 1. {x + 2yy = x^ + 3x^{2y) + 3x{2y)^ + (2y)\ = x^ + 6x*y+l2xy^ + 8y^. Ex. 2. (32r-2i/)3 = {3x)3-3(3x)'(2i/) + 3(3x)(2i/)«-{2i/)3, = 21x^-54x^y+36xy^-Sy^. Ex. 3. (a^-6 + c)3 = (a-^-fe + c)^ = (a + 6)3 + 3(a + 6)»c + 3(a + 6)c« + c3, = a» + 3a*6 + 3a6» + 63 + 3aSc + 6a6c + 36«c + 3oc* + 36c« + c', =a* + b* + c' + Z{a*b + ab* + b*c + bc* + c*a + ca*) + 6abc. EXTRACTION OF ROOTS 223 E5XBRCISB 107 (1 -12. Oral) Find the cube of : 1. -\. 2. -2a. 3. -3a62. 4. — ar^yz". 5. x-\-y. 6. x-y. 7. m-\-n. 8. p-q. 9. x+\. 10. x-\. 11. a^+h. 12. \-a 13. x+3. 14. 2x-y. 15. 2a +36. 16. l~2a. 17. a— 4b. 18. i-a2. 19. a+b—c. 20. a—b—c. 21.* Simplify {a-\-bf+{a-bf and {a+6)3-(a-6)3. 22. Show that {x-\-y)^=x^-\'y^-'r3xy{x-\-y) and write a similar form for (x— ?/)'. 23. Simplify (a+6+c)H(a+6-c)^ 24. Show that {a-b)^+(b-cf+{c-a)^^3ia-b){b-c)(c-a). 25. Show that the difference of the cubes of any two consecutive integers is greater than three times their product by unity. 26. When x=y-\-z, show that x^—y^—z^—3xyz. 27. Two numbers differ by 3. By how much does the difference of their cubes exceed nine times their product ? 28. Three consecutive integers are multiplied together and the middle integer is fidded to the product. Show that the result must be the cube of this middle integer. What is the cube root of 241x242x243+242? 154. Cube Root of a Monomial. The cube root of an^? quantity is one of the three equal factors which were multiplied to produce that quantity. Thus, the cube root of 8 is 2, of o' is a, of 8a;' is 2x, of a* is a^, of 27o»6* is 3a6*. The cube root of any power of a letter is obtained by dividing the index of the power by 3. The symbol indicating cube root is ^ . Thus, -^125=5, i/a*=a', ^8a;V = 2a;*- 224 ALGEBRA 155. Cube Root of a Compound Expression. The cube root of a^-{-2a^b + 3ab^^b^ is a-\-b, and or a^—2a~b^3nb^-b^ is a—b. Therefore, when an expression of four terms is known to be a perfect cube, its cube root can at once be written down by finding the cube roots of its first and last terms. Ex. 1. — The cube root of x^--6x*y-\-\2xy^ — 8y^ is x—2y, since the cube root of x' is z and of — Sy^ is — 2y. Ex. 2.— The cube root of a^-2a*b + lab*- ^^b^ is evidently a-^b. In the cube of a-\-b, the second term is 3a^b. After finding the first term a of the cube root, we might have found the second term of the root by dividing Sa^b by Sa^, that is, by three times the square of the term already found. Thus, the second term of the cube root in Ex. 1 is — 6x*?/-^3x* or —2y, and in Ex. 2 is — 2a*6H-3a' or -§6. Here three times the square of the first term of the root is the trial divisor, corresponding to twice the first term in finding the square root. Ex. 3. — Find the cube root of 8x«+12x5-30x4— 35a;3+45x2+27a;— 27. The first term in the root is 2x* and the last is — 3. The trial divisor for finding the second term of the root is 3(2x*^ or I2x^ .". the second term of the root is 12x^-1- 12x* or x. .". the cube root is 2x*+s;— 3. It is thus seen that it is easier to find cube root by inspection than to find square root, as in finding cube root there is no ambiguity as to the sign of the last term in the root. 156. Higher Roots. Since (x^)^=x*, we may find the fourth root by taking the square root and then the square root of the result. EXTRACTTON OF ROOTS 225 Also, since {x^)^=z^ and {x^)^=x^, we can find the sixth root by taking the square root of the cube root, or the cube root of the square loot. Thus, the square root of x*+8x^ + 24:X* + 32x+16 is x* + 4x + 4, therefore the fourth root is x-\-2. The cube root of x^-Qx^+\5x* — 20x^+l5x* — Qx+\ is x*-2z+l, therefore the sixth root is x— 1. BXE3RCISE 108 (1-15, Orali State the cube root of : 1. -64. 2. 27a3. 3. -\25a%^. 4. -8(a-6)3 5. x3+3a;24-3a;+l. 6. x^-'^xh^+Zxy'^-y^. 7. a3+6a2+12a+8. 8. 9.x^-\2x^+Qx-\. 9. a;V+3xV+3x2/+l. 10. 64a3-144a2+108a-27. 11. 125x3-75x2+ 15x-l. 12. 27x3-27x22/+9x?/2-j/3. »« - ^ 15. m3 - 9w + — - — ' . "^ 16. :r- - 6x< + 12x2m3 _ gw*. f. ' 17.* In finding the cube root of x«+3x5+6x*+7x3+6x2-t-3x+l, what is the first term in the root ? What is the last term ? What is the trial divisor for finding the second term ? What is the cube root ? Check by substituting x=l. ^ ^ ' t. ^ Find the cube root and check : ^ ^*l u^^Sim v^lS. l-6x+21x2-44x3+63x*-54x5+27x«. I-Z^^-Ja..* »** -n a;3 x2 ^ „ 18 27 , 27 .^ - / ^l Ik^im a9. 2-,-3+2x-7 + ---. + ^- -f ^-iZ ^^ t^ 20. 27a«-108a» + 171a*-136a=»+57a'=-12a+l. ^q^" VQ. -♦/ 21. (I + 3x2)2-x2(3+x2)2. ^^''•^itOiay 22. For what value of x will x3+3cx2+2c2x+5c3 be a perfect cube ? 23. Find the fourth root of X*— 4x^+6x2— 4x+l, 1 (f)i '^i.i^-.'i.'n '^ 226 ALGEBRA 24. Find the fourth root of a« — 12a3-)-54a*— 108a + 81 25. Find the sixth root of x«-12x* + 60x*- 160^3+240x2- 192x4-64. EXERCISE 109 (Review of Chapter XVII) Find the square root of : 1.* 9z*-24x»t/ + 28x*2/»-16xt/»4-4t/«. 2. x« + 4xS-2x«-10x'+I3x*-6x+l. 3. xi« + 6xi'' + 5x*-8x«+16x*-8x* + 4. 4. Jx«-Jx»+Vx*-5x+l. 6. 12a3x-26a*x* + 25x« + 9a«-20ax3. 6. 4x»(7-f x* + 3a) + (3a+7)». 7. (x* + 5x + 6)(x« + 7x+12)(x* + 6x+8). 8. (2x'-x-3)(x*-4x-5)(2x*-13x+15). 9. 4x« - 20x» + 33x» - 32x + 34 - — + -^ . XX* What is the cube root of : ^_^ 10. 27-136x+225x»-125x». -• t, * J 11. 8x«-12x'>+]8x«-13x»-f 9x«-3x+l. '^* 'I** "*" 12. (a-fe)3 + 36(fc-»)*-f 3fe»(«-*) + i». 13. Find to three terms, the square roots of : l-2x, 1-a, 4 + x. 14. Find the value of y for which x* — 2(a — y)x + y* is a complete square and prove by trial that your result is correct. 16. The first two terms of a perfect square are 49x* — 28x*, and the last two are -|-6x+|. What must the square root be ? 16. Prove that the product of any four consecutive integers increased by unity is a -perfect square. 17. Find the square root of a* + 4a*+6o* + 4a+ 1 and deduce the square root of 14,641. 18. By finding the cube root, simplify {a + b)' + 3(a + b)*(a-b} + 3(a + b){a-b)' + {a-b)\ t». If a = fc4- 1, show tiiat a> fc»- l = 3a6. 3; z EXTRACTION OF ROOTS 227 20. Show that the product of any four consecutive even integers increased by 16 is a perfect square. How might the result be deduced from No. 16 ? 21. By inspection, find the values of (a-6)*+(ft-c)* + (c-a)* + 2(a-6)(fe-c) + 2(6-c)(c-a) + 2(c-a)(a-6), {2x-y)^-Z{2x-y)^{2x + y) + Z(2x-y){2x+y)*-{2x + y)\ 22. To the square of the double product of any two consecutive integers, add the square of tlieir sum. Prove that the result is always a perfect square. 23. Express in symbols : The difference of the cubes of any two numbers exceeds the cube of their difference by three times theit product multiplied by their difference. Prove that this is true. 24. The expression 8x9-36a:8 + 66x'-87a;«+105x*-87x* + 61x»-42x«+12x-8 is a perfect cube. Find its cube root by getting two terms from the first two terms of the expression and the other two from the last two terms. Check when x=I. 25. What number must be added to the product of any four consecutive odd integers so that the sum may be a perfect square ? 26. Show that the sum of the cubes of three consecutive integers exceeds three times their product by nine times the middle integer. 27. Find the cube root of (4x-l)» + (2x-3)3 + 6(4x-l)(2x-3)(3x-2). [Note that it is of the form a^ ^b^ -\-Zab{a-\-b).'\ 28. If 4x*+ 12x^4- 5x*—2x* are the first four terms of an exact square, find the remaining three terms. ^y^<^-^ Q 2 CHAPTER XVIII QUADRATIC SURDS 157. Surd. When the root of a number cannot be exactly found, that root is called a surd. Thus, we cannot find exactly the number whose square is equal to 2, and we represent the number by the symbol a/2 and we call V2 a surd. If no surd appears in any quantity, it is called a rational quantity. By the arithmetical process of extracting the square root of 2, we can obtain the value of V 2 to as many decimal places as we pleaae, but its exact value can not be found. To four decimal places the value of V2 is 1-4142. Find the square of 1-4142 by multiplication and see how closely it approximates to 2. We can find geometrically a line whose length is V2 units. In this square, whose side is 1 unit, draw the diagonal BD. Then, from geometry, we know that :. BD*=l*+\* = 2, :. BD =V2. On squared paper mark the corners of a square whose side is 10 \inits. Measure the diagonal and thus estimate as closely as you can the value of V 2. Make a diagram like this to show how to represent graphically lines whose lengths are V^, VS, vi, V5, etc. Take the unit line 1 inch in length. What test have you of the accuracy of your drawing ? 158. Quadratic surd. A surd hke V2 in which the square root is to be found is called a quadratic surd. In this Chapter quadratic surds only are considered. QUADRATIC 3URD9 228' 159. Multiplication of Simple Surds. Since \/2 represents a quantity whose square is 2, .-. \/2x V'2=2=V4, also ^4x^9 = ^/36, because 2x3=6. Similarly, we might expect that -v/2x V3 = V6. That this is true may be shown by finding the square of y/2 x V3. (\/2xv'3)2=V2xV3x\/2xV3, [Just as {ab)'^=a .b .a .b.] = \/2xV2x VSxVS, =2x3=6. .-. a/2x\/3=\/6. Similarly, A/3xV'5=Vi5, and y/a x Vb= Voft. Therefore, the product of the square roots of two numbers is equal to the square root of the product of the numbers. Since Vab=VaxVb, .'. \/f2=V4x a/3=2a/3, and V'50=\/25x V2=5\/2 ; VT8a^=V9a^xV2a=3aV2a. Thus, we see that if there is a square factor under the radical sign, that factor may be removed if its square root be taken. Conversely, 5^3= a/25 x V3= VTS, aVb=Va^xVb=Va*b, axViny= Va*x* x Vmy= Va'x*'my. 160. Mixed and Entire Surds. When a surd quantity is the product of a rational quantity and a surd, it is called a mixed surd. If there is no rational factor it is called an entire surd. Thus, 5V3, aVb, {a — b)Vx — y are mixed surds, and V3, V50, Vax-\-b are entire surds. In the preceding article we have shown that a mixed surd can always be changed into an entire surd, and an entire surd can sometimes be changed into a mixed surd. 230 ALGEBRA A surd is said to be in its simplest form when the quantity under the radical sign is integral and contains no squar* factor. Thus, the simplest form of V50 is 5\/2. EXERCISE 110 (1-29. Oral) Find the product of : 1. V2, V3. 2. VS, a/5. 3. V2, Vs. 4. 3\/7, 2Vl. 5. a/3, a/5, a/2. 6. V8, VJ. 7. Vf. Vf. V^ 8. {Vabc)^ Express as entire surds : 9. 2\/3. Ir^ 10. 3\/2. 11. 5a/5. 12. aVb. 13. 3aVl. 14. ' v/i- 15. Wa-b. /a—b - 16. (a+6)x/^^,.,V Simplify, by removing the square factor : r 17. Vs. 18. a/ 12. 19. a/20. 20. \/75. 21. V27. 22. a/56. 23. A/T62. 24. a/2^*. 1 , 25. Vl000a;3. 26. iA/32. 27. A/(a-6)3. 28. -v/o*6. 29. Solve x^=2 ; 3x2=27; ix2=9. 30. Show by squa ring that \/3xa/7 = V''2I and A/a x a/6 X a/c= Vafec. 31. Show that a/8=2a/2, by extracting the square roots of 8 and 2 to three decimal places. 32.* Describe a right-angled triangle whose sides are 2 inches and 3 inches. Express the length of the hypotenuse as a surd. 33. By using a right-angled triangle, how could you find a line whose length is a/ 10 inches ? QUADRATIC SURDS 231 f ^4. If the area of a circle is 66 square inches, find the length of the radius (7r = 3^). 35. The sum of the squares of two surds, one of which is double the other, is 40. Find the surds. 36. The length of the diagonal of a square is 10 inches. Find the length of the side. 37. One side of a rectangle is three times the other and the area is 96 square inches. Find the sides. 161. Like Surds. In the surd quantity 5\/3, 5 is a rational factor and VS is called a surd factor. When aurds, in their simplest form, haye the same surd factor, they are called like surds or similar surds, otherwise they are unlike surds. Thus, 3V2, 5^2, aV2 are like surds. " "^ and 2\/3, 3\/2, jV5 are unlike surds. 162. Addition and Subtraction of Like Surds. Like surds may be added or subtracted, the result being expressed in the form of a surd. Thus, 3V'2 + 5V'2 = 8-\/2, just as 3o+5o=8a. 7V'3-4\/3 = 3\/3, just as 7a-4a=3a. a/76-2v'3 = 5\/3-2v/3 = 3V3. V'50 + V32-\/r8 = 5V2 + 4V2-3\/2 = 6\/2. Che sum or difference of unlike surds can only be indicated. Thus, V2-\-V^ can not be combined into a single surd, but the approximate values oi V2 and V3 may be found and added. Show that V2+v'3 = \/5 is not true, by finding the square roots of 2, 3 and 5 each to two decimals. Is it true that V'i + VQ=VYZ ? BXBRCISE 111 (1-8, Oral) Express as a single surd : 1. 3^2+5^2. 2. 5-\/7-3V7. 3. 2Va+3Va. 4. 2Vx+5Vx-Vx. 6. VS+V2. 6. ^12+^3. 232 ALGEBRA 7. Vis-Vs. "^ V75+Vr2+3\/3. Vi5-V2b+V80. 11 8. V4a+V9a. 10. 2V18+3a/8-5\/2. 12. 2\/63-5\/28+a/7. 14. 10^44-4^99. 13. 4\/f28+4\/56-5Vr62. 15. A/45+V20-\/80+\|r80. 16. \/72+a/98-V128-V32-\/50. Simplify the following and find their numerical values, correct to two decimal places, using the square root table : 17. VT5. 18. V63. 20. VI^-2VI2. 21. VI28-VT62. 19. 2^ V60+\/l5. V56+V'72+V'90. Solve, finding x to three decimal places : 23. x2=37«***'^, 3x2+5=50. 26. 31x2=132. 27. i(3x2-ll)=53. 25. 28. 29. The area of a circle is 176 square inches Square Roots of Numbers from 1 to 50 ix2-4 = 19. ^x2=Jx2_47. Find its radius. n Vn n 11 3-3166 n 21 4-5826 n Vn n 41 y^ 1 10000 31 5-5678 6-4031 2 1-4142 12 3-4641 22 : 4-6904 32 5-6569 42 6-4807 3 1-7321 13 3-6056 23 ' 4-7958 33 5-7446 43 6-5574 4 20000 14 3-7417 24 4-8990 34 5-83i0 44 6-6332 5 2-2361 15 3-8730 25 50000 35 5-9161 45 6-7082 6 2-4495 16 4-0000 26 5-0990 36 6-0000 46 6-7823 7 2-6458 17 41231 27 1 5-1962 37 608^ 47 6-8557 8 2-8284 18 4-2426 28 5-2915 38 6- 1644 48 6-9282 9 3-0000 19 4-3589 29 j 5-3852 39 6-2450 49 70000 10 31623 20 4-4721 30 5-4772 40 6-3246 50 70711 t68. Multiplication of Surds. 3 V2 X 4a/3=3 X V2 X 4 X V3, =3x4xV2xV3, = 12a/6. QUADRATIC HvltDS 233 It is thus seen that the product of two surds is found by multiplying the product of the rational factors by the product of the surd factors. 5V3x2V3=10.3=30, also ay/cxhVc=abc. It, therefore, follows that the product of two lik^ surds i' always a rational quantity. Ex. 1.— Multiply VSO by Vl5. Here the surds should be simplified before multipl3dng. Since V50 = 5V'2 and V75 = 5V'3, .-. V50X V75 = 5\/2x5V3 = 25v'6. Ex. 2. — Multiply 2-i-2\/3 2 + 2V3 by 3-V2. ^~ ^^ Here the multiplication is performed ma 6-j-6v3 manner similar to the multiplication oi a-\-b — 2v2— 2v6 by x + v- ~ ~ ^ 6 + 6\/3-2\/2-2V6- 164. Conjugate Surds. If we wish to multiply 5V3+2V2 by 5\/3-2\/2, we may follow the same method as in the preceding example. These expressions, however, are seen to be of the same form as a-\-b and a—b, :. (5\/3 + 2V'2)(5V3-2\/2) = (5\/3)2-(2\/2)2=75-8=67. Similarly, (3+ V2)(3- V2) = 9-2 = 7, and (2-Vl0)(2+\/l0) = 4-10=-6. Such surd quantities as these which differ only in the s)gn which connects their terms are called conjugate surds. Note that the product of two conjugate surds is always a rational quaniitv. 234 ALOE BRA E3XERCISB 112 (1-12. Oral) Find the product of : 1. 2V3, 3a/5. 2. 5\/2, 6a/3. 3. aVb, bVa. 4. W2,V3,V5. 5. {2\/3)2, (V2)2. 6. \/2+l, ^2. 7. A/3-1-V5, -\/2. 8. \/a+V'6-l, Vc. 9. V3 + V2, V3-V2. 10. Vro- 3, VIO+3. 11. Vx—Vy,Vx+Vy. 12. 2\/2-|-\/3, 2v'2— VS. ^?3a/6, 4V2. 14. 3V^, 4\/7,J\/2. 15. {\/3+a/2)2. 16. (2\/6-V7)'- 17. (3\/2-|-2V'3)2. 18. (Va+\^)2. 19. 4-1-3V2, 5-3A/2. 20. 3\/2 + 2V3, 5^2-3^3. 21. ZV5-W2, 2\/5+3"v/2. 22. 3Va~2Vb, 2Va-3Vb. 23. V5+V3+V2, ^5+^3-^2. ^Sk' V7+2V2-V3, V7-2V2+V3. 25. Va+b—3, Va+b-]-2. 26. Va+Va— 1, Va— Va— 1. 27. (a/3+\/2-|-1)2. 28. (V'5+2\/2-\/3)^- 29-. (Va+b+Va^)^. i^-^J! (3V'^^-2Vx+y)^- Simplify : 31. (6-2a/3)(6+2\/3)-(5-V2)(5+v/2). .-/ ^^,.32f"(\/3-\/2+l)2+{A/3-|-V2-l)2. ^8^r^50- VI8+ \/72+ V32) X iVS. ^3^/2(4\/3-|-3\/2)(3\/3-2\/2) + (5\/2"-3\/3)(4\/2+2\/3). y«5. ( V3+ \/2)(2\/3 - \/2)( \/3-2\/2)( V3-3V2). 36. By squaring VlO-f Vs and Vs+Vl, find whicii is the greater 37. The product of 5\/3-f SV? and 3^3— V? lies between what two consecutive integers ? 38. Find the area of a rectangle whose sides are 5+ V'2 and 10— 2^? taches. QUADRATIC SURDS 235 39. The sides of a right-angled triangle are 7-(-4V2 and 7 — 4v'2 inches. Find the hypotenuse. 40. The base of a triangle is 2^3 + 3^2 inches and the altitude is 3V3+2\/2 inches. Find the area to two decimal places. 165. Division of Surds. • Inh Since VaxVb=V^, :. Vab^Va=J-'=Vb. y a Similarly, Va ~ Vh = —7= = \/ - , Vh ^ h and 3\/l5-f2V5=|V3. Ex. 1. — Find the numerical value of V5-f- V2 or —p^ • V2 (1) We might find the square roots of 5 and 2 and perform the required division. VSh- ^2 = 2-236-:- 1-414= 1-581 (2) V'5-^V2=\/|=V'275= 1-581. ,_, V5 •\/5xV2 \/lO 3-162_,_-- Here the third method is at once seen to be s.'^pler than either of the others. Vs . VlO In (3) we changed — -^ into , that is, we made the ^ , V2 2 denominator a rational quantity. This operation is called rationalizing the denominator. Ex. 2.— Find the value of ^^. if ^2=14142. V2 Here, instead of dividing 1 by 1-4142, we first rationalize the denominator. ™ 1 lxV2 a/2 1-4142 _-_, Then —^ = —^ y^ = -jr- = — ^ — = -7071. \/2 •v/2xV2 2 2 Ex. 3.— Divide 6^8 by 10a/27. 6V8 6x2\/2 2^2 2xV2xV3 2\/6 2x2-4496 mV2i JOxsVa sVa sxVsxVs is 15 = -32W 236 ALGEBRA hx. 4. — Rationalize the denominator of -• We have already seen that the denominator will be rational if we multiply it by its conjugate 3— V 5. 2+\/5 _ C2+V'5)(3- \/5) ^ l + Vh ^ l + Vl 3+A/5 ~^4-a/5)(3-V5) ~ 9-5 ~ * Ex. 5.— Divide 5 + 2^3 by 7-4\/3. 5 I 2\/3 Write the quotient in the fractional form ,_. rationalize the 7-4\/3 . denominator and simplify. EXERCISE 113 (1 Divide : 12. Oral) 1. 3a/27-^\/3. 2. V12^a/3. 3. \/72^3\/8. 4. VofccH-Va. 6. vlS + Vl^by Vs. 6. Voft+Voc Rationalize the denominator of : 2 10 ^- V3" *• V5 9. Vb -v/6 12. (§>7i: V5 V3' I v2-r ^.. . 14 i! 15 V3 + V2 4V3 ' 3V2-2\/3 ' v'3-V2' 16 V^o 17 5 V3-3A/5 ^^ V7 + V2 Va+V6 " V5-VZ ' ' 9+2\/Ii Find the value to three decimal places, using the table : 1 15 /-^ 2V3 19. -7=- 20. -7= • (21) f-L^. V3 VI8 V^ 3V2 1 17 Vi-Vb 20 — . 23 = • **4 — — - "* V3+V2 ' 3^7+2^3 " ' V7+V5 25. V3-r\/2. 26. 2\/63-^3V'35. 27. l-r(7+4V3). QUADRATIC SURDS JSSTi Solve, giving the value of x to two decimal places, using the table ; 28. a;\/2=3. 29. xVz=V2. 30. a;V'3=\/2 + l. 31. x\/2-xV2=\. 32. xV5-5=2x-y/5. 33. x2(\/3-l)==2(\/3+l). 34. The area of a triangle is 2 square feet. The altitude is V6-\-VZ feet. Find the base to three decimals. 35. Simplify 2+VlO 4\/2+\/20-\/18-a/5 166. Surd Equations. A surd equation is one in which the unknown quantity is found under the root sign, in one or more of the terms. Thus, V'a;+7 = 4, Vx+Vx— 5 = 5, are surd equations. Ex. 1.— Solve Vx^=2. Square both sides, a;— 3 = 4, .'. x=7. Verification : Vx^ = VT^ = Vi = 2. Ex. 2.— Solve V5^^-2a/^3=0. Transpose 2\/x+3, \/5x—\ = 2^/x-\-%. Squaring, 5x— l = 4a;+12, .-. x=13. Verification : -v/5a;— 1 — 2 Vx4- 3= \/64— 2-^/^6 = 8— 8 = 0. Note that in verifying we have taken the positive square root only, as defined in art. 63. E3XBRCISB 114 (1-8, Oral) Solve and verify : 1. 2Vx=Q. 2. Vx— 5=4. 3. Q—\/x=l. 4. \/x+2=4. 5. Wx=V20. 6, Vx— 6=a. 7. m+V'i=n. 8. 7— Vx— 4=3. 9. Vx2+9=9-x. 10. Vx2+llx+3=x+5= ' . 11. V9x2-llx-5=3x-2. CBL 2x-\/4x2-10x+4=4. @ 2a+Vz+a^=b+a. 44^, V(x-o)»+2a6+6»=*-a+6, «38 ALGEBRA EXERCISE 116 (Review of Chapter XVIII) Simplify : 1* a/8+\/T8 + a/98. 2. V500+ V80- V20. 3. 5a/3 + 3V'27-V'48. 4. (4V5+ VT8)(4\/5- VFS). 6. (6\/6-6)(6V'6 + 5). 6. (V6+V2+2)(\/6-\/2-2). 7. (\/8+V2-2)». 8. (V'3-2V'2-1)». 9. 5\/27-^6\/75. 10. (\/5-2)^(\/5 + 2). LI. (^125+^/45)^^320. 12. (5+^/3)(5-V3)-^(^/^3-\/2), 13. Multiply 3\/8 + 2v'3-V'2 by 2\/8- V'3 + 4V'2. 14. By how much does the square of VS -\ — exceed the square of a/2 4- 4^ ^ V2 15. Show by multiplication that the value of Vs lies between V732 and 1-733. Which of these is the closer approximation to VS 7 16. Which is the greater, VU+VS or VB+Vro? 17. The product of 3\/2-2\/3 and 2\/3-V2 lies between what two consecutive integers ? 18. Rationalize the denominators of : 4 3 a/2 3 /5 ^g;g 2a/6-2 \/2' 2V3' 2V 6' ■ ' 3\/3 + \/2' Solve and verify : 19. \/x+3 = 4. 20. A/3a^ = 2\/a;-2. 21. y/x*-5+\=x. 22. \/2a;+7 = 3\/«. 23. a/x»-5x+11=x+2. ,24^ \/x»-2=l-x. 25. Using the table, solve: x» = 75, x* = 63, Jx* = 49, x-v/3=V'5, T\/2+l = \/3. ^re. Find to three decimal places the values of : 2 1 1 2A/rO-A/5 3V2-2 Ve' VB' V2+r a/Io+Vs ' 4V2+r QUADRATIC SURDS 239 ^2f\ Find the value of (2V2+V3){3V2- \/3)(3 V3- V2). 28. If the sides of a right-angled triangle are Vs+l and V3— 1, what is the length of the hypotenuse ? oo^ c!- r. ^5-1 V5-3 , V 3+V2 V3-V2 I 28C Simplify -7= jz: and —j:z 7^ 7= ~ • ^^ Vo-2 \/o + 3 V3-V2 V3+A/2 30. Find the value to two decimal places of x+y x-y ^henx=2 + V'3, y = 2-V3. x—y x+y 4.,afr^ Multiply 2\/30-3V'5 + 5\/3 by V3 + 2\/2-V5. t,3gf^ Multiply ^7 + 2^6 by \/7-2V6. 33. The area of a rectangle is ISVlO — 25 and one side is 3V5— V2- Find the other side to two decimal places. CHAPTER XIX QUADRATIC EQUATIONS 167. A quadratic equation has already "been defined in art. 104. In the same article we considered the method of solving some of the simpler forms of it. Quadratic equations frequenlly occur in the solution of problems as shown in the following examples. Ex. 1. — Find two consecutive numbers whose product is 462. Let the numbers be x and x-\-\. a;(a;+l) = 462, .-. x»4-a;-462 = 0. Ex. 2. — The length of a rectangle is 10 feet more than the width and the area is 875 square feet. Find the dimensions. Let x = the number of feet in the width, .". a:+ 10 = the number of feet in the length, x(x+10) = 875, .-. a;«+l Ox -875 = 0. Ex. 3. — Divide 20 into two parts so that the sum of their squares may be 36 more than twice their product. Let a; = one part, 20 — x = the other part, x» + (20-a;)*=2a;(20-a;) + 36, .-. x« + 400-40x + x» = 40x-2x* + 36, 4x»- 80x4-364 = 0, x»-20a; + 91 = 0. J40 QDADRATIC EQUATIONS Ul exercise: 116 Represent the number to be found by x and obtain, in its simplest torm, the quadratic equation which must be solved in each of the following : 1.* The sum of a number and its square is 132. Find the number. 2. Find the number which is 156 less than its square. 3. The sum of the squares of three consecutive numbers is 149. Find the middle number. 4. The product of a number and the number increased by 6 is 112. Find the number. 5. The length of a rectangle is 6 feet less than five times the width. The area is 440 square feet. Find the width. 6. The average number of words on each page of a book is 6 more than the number of pages. The total number of words is 9400. Find the number of pages. 7. The area of a rectangle is 88 square inches and the perimeter is 38 inches. Find the length. 168. Standard Form of the Quadratic Equation. Every quadratic equation may be reduced to the form ax^-\-bx-\-c^O, in vv'hich a, b and c are any known numbers, except that a 'jan not be zero. The term not containing x is called the absolute term. It is frequently necessary to simplify equations to bring them to the standard form, and thus determine if they are quadratic equations. Ex. 1. (a;+l)(2a;+3)=4x2-22, 2x* + 5x + 3 = 4a;«-22, .-. -2a;* + 5x + 25 = 0, 2a;»-5x-25 = 0. Here the equation is seen to be a quadratic. The coefficient of x* is 2.. of a; is —5 and the absolute term is —25. Or, o = 2, 6= — 5, c=-25. 242 ALOE BR A Ex. 2. -j- + = 1, 4 X :. 7*« + 4(a;-7) = 4x, 7x»-28 = 0, x»-4 = 0. Here a=l, 6 = 0, c = --4. 2x a: 4-1 .-. 2x(x+2) + (x+l)(x-l) = 3(a;-l)(ar+2), 2x* + 4x+x*-l-=3x« + 3x-6, x+5 = 0. Here a = 0, 6=1, c = 5, and the equation is not a quadratic, since the coefficient of x* is zero EXERCISE 117 Reduce to the standard form and state the values of a, b and c, in which a is always positive : 1.* 6x2=x+22. 3. 19x=15-8x«. 6. — -^ — = 5x. 2. 25x=6x24-21. 4. 2=llx-12x2. 6. " + x = 2- 4 8. (3x-5)(2x-5)=x2+2x-3. ^. 3g-8 ^ 5x-2 ^ 5_ _ 3 x-2 " x+5 ' *^ ■ x-1 x-|-2~a;' =^+l + J^=4. 12. 2x a:-3_^ w/ x+2 x-l t/^ x-3 X 169. Solution by Factoring. (1) When the absolute term is zero, the equation ca| always be solved by factoring. QUADRATIC EQUATIONS 243 Ex. 1.— Solve 2x^—3x=0. a;(2x-3) = 0, x=0 or 2x-3 =0, x = or f. Verify both roots. Ex. 2.— Solve ax^+bz=0. x{ax + b) = 0, .'. x = or ax + b = 0, b .'. x = or • a (2) \Mien the middle term is zero, the equation can always be solved b}' factoring, or bj' extracting the square root. Ex.— Solve 3a;2— 27=0. 3(z-3)(x + 3) = 0, .*. a;-3 = or x-f3 = 0, x=±3. Or thus, 3x»-27 = 0, .-. x* = 9, :. x=±3. (3) The equation is a complete quadratic when none of the coefficients a, b, c is zero. If the quadratic expression, ax^-\-bx-\-c, can be factored b}' any of the methods previously given, the solution is then easily effected. Ex. 1.— Solve 3x2-lla;=14. 3x«-lla;-14 = 0, .-. (a;+l)(3a;-14) = 0, a; = — 1 or V . Verify both of these roots. Ex. 2. — Solve x^—mx-\-nx—mn^=0. t(x — 7n)-(-n(x — m) = 0, .'. (x — m)(x + n) = 0, x = m or — n, E 2 244 ALGEBRA EXERCISE 118 1-12. Solve the equations in the preceding exercise and verify. 13-19. Solve the problems in the first exercise in this Chapter. (Verify the results.) Solve by factoring and verify : ^^. a:^— mx— 6/7j2=rO. ^^. x^— ax— 6x+a6==0. 24. x2-L2x(a+6)4-4a6=0. 25. 2ax2+aa;— 2x=l. 26. (x— a)(x-6)=a6. 27. x^— a2=(x-a)(6+c). 170. Consider the problem : Find two numbers whose sum is 100 and whose product is 2491. Let a:=one number, 100— x=the other number, a;(100-x) = 2491, .-. x2- 100^+2491=0. To .solve this equation by the preceding method, we must find two factors of 2491 whose sum is 100, but this is exactly what the problem requires us to find. The necessity is therefore seen for another method of solving the quadratic equation when the factors of the quadratic expression cannot be obtained readily by inspection. 171. Solution by Completing the Square. We know that (x-|-a)2=x2+2ax-i-a2, the middle term being twice the product of X and a. If the first two terms of a square are x^A^lax, we know that it must be the square of x+a, and, therefore, a- must be added to x^-\-2ax to make a complete square. What is the area of the stiaded portion in * " the diagram ? Similarly, x*-i-4x must be the first two terms in the square of x+2. To make x'-(-4x a complete square we must add 2* or 4. Also, x* — 8x are the first two terms in the square of x — 4, and, therefore, 4' or 16 must be added. asc m x' ax QUADRATIC EQUATIONS 245 To complete the square, it is seen that the quantity to be added is the square of half of the coelTicient of jc. Ex. 1.— Factor a;2^6a:— 40. Add 9 to x'-\-6x to make a complete square. Then x* + 6a;-40 = a;»+6a;+9-9-40, =a;*+6a;+9-49, = (*+3)*-7», = (a;+3 + 7)(a; + 3-7), = (x+10)(x-4). Ex. 2.— Factor x^+5xS0Q. Add (I)- or ^^- to x*-\-5x to complete the square. Then x* + 5x-806=x^+5x + ^-\^--S0Q, = (a;+|)«_(V-)*, = (a:+§ + V^)(x+§-V). = (x+31)(x-26). Ex. 3.— Solve a;2-100a;+2491=0. Add 50* or 2500 to complete the square. x"- 100x4-2500-2500 + 2491 = 0, x2-100x + 2500- 9 = 0, (x-50)2-32 = 0, .-. (x-50 + 3)(x-50-3) = 0, {x-47)(x-53) = 0, x-47 = or x-53 = 0, x=47 or 53. The solution might be contracted by writing it in the following form : x»-100x+2491 = 0. Transpose the absolute term, .'. x^— 100x= — 2491. Add 2500 to each side, /. x* - 100x + 2500= -2491 + 2500=9. Take the square root, /. x — 50=±3, x=50±3, = 53 or 47. 246 ALGEBRA Here the solution depends upon the same principle, but assumes a simpler form. It is thus seen that we effect the solution of a quadratic equation by finding and solving the two simple equations of which it is composed. Thus by the first method of solving x2— 100x+2491=0, we obtained the two simple equations x— 47 = and x— 53=0, and by the second x— 50=3, and x— 50= — 3. Ex. 4.— Solve 3.r2+x=10. Divide by 3 to make the first term a square, Add (J)» to each side, .-. x«+ ia; + 5V = V+irs = Vv- Take the square root, .". 2;+i = + Y. x=±V-— 8 = 3 or —2. Verify both of these roots. The steps in this method are : 1. Reduce the equation to the standard form and remove the absolute term to the right. 2. Divide by the coefficient of x^ if not iinity. 3. Complete the square by adding to each side the square of half the coefficient of x. 4. Take the sqvxire root of each side. 5. Solve the resulting simple equations. EXBRCISE 119 (1-8, Oral) What must be added to each of the following to make a comolete square ? 1. x^+2x. 2. a;2-4x. 3. x^+\Ox. 4. x^-Ur. 5 x^+3x. 6. x2— 5x. 7. x2-|-4ax. 8. x^— .J.c Factor, by making the difference of squares, and verify : ^. x^+'Lt—n. JO. x2— 54x-(-713. -fT. x^— 2x— 899 (l^ x»-x-1640. ^^ x'-i^x+^. (l4r'3x«-|-16x-99. QUADRATIC EQUATIONS 247 Use the method of completing the square to solve the following and verify the roots : ^ 2;2^8x=9. ^. x2— 6x=7. Qi x^— 10x+9=0. ^. x2-9x+18=0. & x2+7x4-10=0. €^ x2-x=2. 21. 2x2-3x=2. 22. 2x2+x=1081. 23. 6x2+5x=6. 24. If x2+x=l^, find the values of x + - • 172. Equations with Irrational Roots. In all the quadratic equations we have solved, we found that when we had completed the square on the left side, the quantity on the right was also a square. This would not always be the case. Ex. 1.— Solve a;2-6a;-l=0. a;«-6x=l, .-. x»-6x+9=10, x-3=±v'IO, x = 3±v'ro. The two roots are 3 + VTO and 3-VTO. We might go a step further and substitute for VlO its approximate value 3- 16. The two roots would then be 3±316 = 616 or -16. If we substitute- either of these values for x in x' — 6x— 1, the result will not be exactly 0, as we might expect, because \^10 is not exactly 3-16, but the difference between and the value found for x' — 6x— 1 will be very small. Ex. 2.— Solve 2x^+x=2. x» + ix=l, x + i=±\/H=±iVl7. x=-i±^Vr7. The two roots are -\-\-kV\l, -\-kVV1, or '781, -1-281, on substituting V 17 = 4- 123. 248 ALOE BRA 173. Inadmissible Solutions of Problems. When a problem is solved by means of a quadratic equation, it does not follow that the two roots of the equation wull furnish two admissible solutions of the problem. Ex. — A man walked 25 miles. If his rate had been one mile per hour faster he would have completed the journey in 1^ hours less. What was his rate ? Let his rate be x miles per hour. 25 The time taken to walk 25 miles = — hr. X 25 At the supposed rate his time- = r hr. x-\-l 25 _ 25 •• X " x+l~ ^' Simplifying, a;* + a;— 20 = 0, Solving, x=4 or —6. Therefore his rate was 4 miles per hour, the other root giving a solution which is inadmissible BXE3RCISE 120 Solve, finding the roots approximately to three decimal places, using the table : 1.* x^—4x=\. ^: a;2— 10x+17 = 0. -3. «H2x-6=0. k. :c2+8a;=19. 6. x(a;+3)=J. ^6. 2a;2+3a;-4=0. Solve, expressing the roots in the surd form : y 7. a;2— 6x=2. 8. x'^^9,x=\\. 9. 4a;2— 4a;=7. 10. 4x2-8x=37. 11. 3x2-5a;-ll=0. <12. \x^^\x=\. The following problems reduce to quadratic equations. In solving the equations factor by inspection where possible and verify the results. 13, The sum of two numbers is 11 and their product is 30. Find the numbers. 14. The sum of the squares of two consecutive numbers is 85. Find the numbers. QUADRATIC EQUATIONS 249 15. The difference between the sides of a rectangle is 13 inches and the area is 300 square inches. Find the sides. I 16. Find two consecutive numbers such that the square of their 'sum exceeds the sum of their squares by 220. 17. A merchant bought silk for $54. The number of cents in the price per yard exceeded the number of yards by 30. Find the number of yards. 18. The area of a rectangular field is 9 acres and the length is 18 rods more than the width. Find the length. 19. The three sides of a right-angled triangle are consecutive integers. Find the sides. 20. How can you form 730 men into two solid squares so that the front of one will contain 4 men more than the front of the other ? 21. The owner of a rectangular lot 12 rods by 5 rods wishes to double the size of the lot by increasing the length and width by the same amount. What should the increase be ? 22. If x-\-2 men in x+5 days do five times as much work as a:+l men in x— 1 days, find x. 23. A rectangular mirror 18 inches by 12 inches is to be surrounded by a frame of uniform width whose area is equal to that of the mirror. Find the width of the frame. 24. What must be the radius of a circle in order that a circle with a radius 3. inches less may be | as large ? > 25. One side of a right triangle is 10 less than the hypotenuse afld the other is 5 less. Find the sides. 26. A man spends $90 for coal, and finds that when the price is increased $1-50 per ton he will get 3 tons less for the same money. What was the price per ton ? 27. A man bought a number of articles for $200. He kept 5 and sold the remainder for $180, gaining $2 on each. How many did he huy? 28. The sum of the two digits of a number is 9. The sum of the squares of the digits is f of the number. Find the number. 29. A number of cattle cost $400, but 2 having died the rest a'vera^ed $10 T>8r head more Find the number bou|;ht 250 ALGEBRA 30. How much must be added to the length of a rectangle 8 inches by 6 inches in order to increase the diagonal by 2 inches ? 31. In the figure, the rectangle AO . 05= rect- angle CO . OD. (1) If .40=16, B0^3, C0= 8, find OD. (2) If ^0=10, £0=4, CD=13, find OD. 32. In the figure, when OA is a tangent to the circle, OA^=^OC .OD. (1) If 0C= 4, CD= 5, find OA. (2) If 0.4= 8, 0Z)= 10, find 00. (3) If 0.4 = 15, 0Z)=16, find OD. 33. I sold an article for $56 and gained a per cent, equal to the cost in dollars. What was the cost ? 34. The denominator of a fraction exceeds the numerator by 3. If 4 is added to each term the resulting fraction is f of the original fraction. Find the fraction. 35. An open box containing 432 cubic inches is to be made from a square piece of tin by cutting out a 3 inch square from each corner and turning up the sides. How large a piece of tin must be used ? 36. A and B can together do a piece of work in 14| days, and A alone can do it in 12 days less than B. Find the time in which /I could do it alone. BXERCISE 121 (Review of Chapter XIX) 1. What is a quadratic equation ? 2. Is (x+l)(a;— 2)(a;+3) = {a; — 4)(x— l)(a; + 7) a quadratic equation? Solve it. 3. The sum of a positive number and its square is 4. Find the number to two decimal placas. ^4. SolvegA_+gA_ = 3;-L-^=l. 5. If x*y* — 6xy — l = 0, what are the values of xy 1 6. Are x = 4 and x'=16 equivalent equations, that is, have they the same roots 7 C9 Solve x'— xy+y* = 39, when y=7. 3t -r * QUADRATIC EQUATIONS 251 8. Divide 14 into two parts so that the sum of their squares may be greater than twice their product by 4. 9. If (a;-2)(x-3) = 7(x-3), does it follow that x-2 = 7 ? What is the proper conclusion ? 10. The distance {s) in feet that a body falls from rest in t seconds is given by the formula s=l6lt*. How long will it take a body to fall 6440 feet ? ir.i Solve 3x « - 4x - 1 = 0. Ten times a number is 24 greater then the square of the number. Does this condition determine the number definitely ? .„ „ , x+2 7 4-a; 13. Solve— ^ =3 + ^^. r ^R. Find two consecutive odd numbers whose product is 399. 15. Solve(2a; + 3)«-2(22; + 3) = 35. 16. The units digit of a number is the square of the tens digit and the sum of the digits is 12. Find the number. ._ „ , a;+10 10 11 7 15 17. Solve — !— = = "E- ; — r-5 5 = 5 • X— 5 X 6 x+5 a;— 3 3 18. If a train travelled 10 miles per hour faster it would require 2 hours less to travel 315 miles. Find the rate. <^. Solve (3j;-7)(2j;-9) — (5a;-12)(a;-6) = (x-2)(2a;-3). 20.* Find, to three decimal places, the positive number which is 'ess than its square by unity. ^yJH*-. If 4a;« — 3xy^-y'=14, find x if j/ = x + 3. ■"'22. The perimeter of a rectangle is 34 feet and the length of the diagonal is 13 feet. Find the sides. 23. Solve x* + (x — 4)* = 40. State the problem, the condition in which is expressed by this equation. 24. A line 20 inches long is divided into two parts, such that the rectangle contained by the parts has an area of 48 square inches more than the square on the shorter part. Find the lengths of the parts. 25. Solve x^ + !/^ = 9, when y = Z—x. 26. The diagonal of a rectangle is 39 feet and the shorter side is i'^ of the longer. Find the area. 27. If 5 is one root of x' — 7x*-f-6x+20 = 0, find the other roota to three decimal places. 252 ALGEBRA 28. Find the price of eggs per dozen when 10 less in a dollar's worth raises the price 4 cents per dozen. 29. The length of a field exceeds its breadth by 30 yards. If the field were square but of the same perimeter, its area would be t^ greater. Find the sides. 30. If 8x — — = 4, find x to three decimal pleices. X 31. The cost of an entertainment was S20. This was to be divided equally among the men present. But four failed to contribute anytjjing, and thereby the cost to ea^h of the others was increased 25 cents. How many men were there ? 32. If a man walked one mile per hour faster he would walk 36 miles in 3 hours less time. What is his rate of walking ? 33. A polygon with n sides has ^n(n— 3) diagonals. If a polygon has 20 diagonals, how many sides has it ? 34. Solve aHa:-a)« = 6*(x + a)*. 35. A can do a piece of work in 10 days less than B. If they work together they can do it in 12 days. In what time could each do it alone ? 36. If x'-L ^ = 81. find the value of x^ and of x. X" * 37. The length of a rectangular field is to the width as 3 to 2 and the area is 5-4 acres. How many rods longer must it be to contain 6 acres ? CHAPTER XX RATIO AND PROPORTION 174. Methods of Comparing Magnitudes. When we wish tc compare two magnitudes, there are two ways in which the comparison may be made. (1) We may determine by how much the one exceeds the other. This result is found by subtraction. (2) We may determine how many times the one contains ;he other. Here the result is found by division. Thus, if one line is 6 inches in length and another is 18 inches, we may say that the second is 12 inches longer than the first, or that the second is three times as long as the first. Neither method of comparison can be used, unless the magnitudes compared are of the same denomination, or can be changed into equivalent magnitudes of the same denomination. Thus, we can compare 3 lb. and 10 lb. ; 2 yd. 1 ft. and 2 ft. 9 in. ; but we can not compare 5 lb. and 4 ft. 175. Ratio. When two magnitudes, of the same kind, are compared by division, the quotient is called the ratio of the magnitudes. Thus, the ratio of 3 to 4 is the same as the quotient of 3-1-4, which is usually written |. The ratio of 3 to 4 is written 3:4, .'. 3:4=3^4 = 1. Similarly, a : b=a^b = -r. 254 ALGEBRA It will thus be seen that all problems in ratio may be considered as problems in fractions. 176. Comparison of Ratios. To compare two ratios we simply compare the fractions to which these ratios are equivalent. Ex. 1. — Which is the greater ratio, 3 : 4 or 7 : 9 ? The problem is at once changed into : " Which is the greater fraction i or i ? " To compare the fractions we reduce them to the same denomination in the forms ^l and §f , and it is seen that the latter is'the greater. We m.ight also compare them by reducing the fractions to equivalent decimals. Ex. 2. — Which is greater, a : a-\-2 or a+1 : a+3. a _ a(o+3) _ a*+3a a+2 ~ (a + 2)(a + 3) "~ (a + 2)(a + 3) " g+l _ (a+l)(a + 2 ) _ a* + 3a+2 a+3 ~ (a+2)(a+3) ~ (a+2)(a+3)" What is the conclusion ? 177. Terms of a Ratio. In the ratio a : b, a and b are called the terms of the ratio, a being called the antecedent and b the consequent. The antecedent corresponds to the numerator of the equivalent fraction, and the consequent to the denominator. _,, o antecedent numerator dividend Thus, r = = -, -. = ,. . • consequent denominator divisor 178. Equal Ratios. Since a ratio is a fraction, all the laws which we have used with fractions may also be used with ratios. Thus, since - = — j- , it follows that a : b=^ma : mb. mb Hence bolh terms of a ratio may be multiplied or divided by the same quantity {zero excepted) without changing the value of the ratio. Thus, 6 : 9 = 2 : 3, i : i = 3 : 2, ^-: - = ^^ : 6» .23 ' 6 o *- RATIO AND PROPORTION 265 EXBRCISB 122 (1-15, Oral) Simplify the following ratios by expressing them as fractions in their lowest terms : 1. 10 : 15. 2. 2J:5. 3. 45 : 63. 4. 15 : 10. 5. $2 : S6. 6. $2-50 : $10. 7. 2 ft. : 3 yd. 8. 2 days : 12 hr. 9. 2 ft. 3 in. : 3 ft. 3 in. 10. 24a : 8a. 11. bah : 10a 2. 12. a+b: a^—b\ 13. a-6:a3-fe3. 14. x—y: x—y. 15. 1-1:1 + 1. X X 16. 1 1 _ . 17. X 2+2«2/+2/2 : x^+y\ 18.* If 12 inches=30-48 centimetres, find the ratio of an inch to a centimetre and of a metre to a yard. 19. The edges of two cubes are 2 inches and 3 inches. Find the ratios of their volumes. 20. If 25 francs=$4-80, find the ratio of a franc to a dollar and of a quarter to a franc. * 21. If a metre=39-37 inches, find the ratio of a kilometre to a mile. 22. Which is the greater 2 : 3 or 4 : 5, 15 : 37 or 11 : 27, a : o+2 or a+3: 0+5 ? 23. Arrange in descending order of magnitude: 2: 3f 3: 5, 11 : 15, 13: 18. 24. What is the effect of adding the same number 5 to both terms of the ratio 7 : 15 ? What is the effect of subtracting 5 from each term? 25. What is the effect of adding 5 to each term of 15 : 7 ? Of subtracting 5 ? Compare your results with the results of Ex. 24. 26. Separate 360 into three parts which are in the ratio 2:3:4. (Let the parts be tx, 3x, 4a;.) 27. Divide 165 into two parts in the ratio of 2 : 3 ; 510 in the ratio 3:7; 36 in the ratio 1^:2^. Q Soieet 256 ALOE BRA f28^ When a sum of money is divided in the ratio 1 : 2, the smaller part is 820 more than when it is divided in the ratio 2 : 7. Find the 3um. (_29. What number must be added to both terms of | to make it equal to i ? (*Ji-^ (30^ What number subtracted from each terra of 7 : 10 will produce 13: 19 ? 31. What must be added to each term of a : b to produce c : d 1 What is the conclusion when c^=d^. 32. If a is a positive number which is the greater-ratio, l+2a 14-3a , or 7 l+3a l+4a The rate of one train is 30 miles per hour and of another is SoTeet per second. What is the ratio of their rates ? 34. Divide a line a inches long into two parts whose lengths are in the ratio b : c. Ji^. ^'s income : 5's income=3 : 4, and A'^ expenditure : 5's expenditure =5 : 6. If ^ spends all his income, what per cent, of his income does B save ? ^^^. Divide 8315 among A, B and C, so that A's, share will be to 5's as 3 : 4, and £'s to (73 as 5 : 7. 37. A line is divided into two parts in the ratio of 5 : 7 and into two parts in the ratio 3:5. If the distance between the points of division is 1 inch, find the length of the line. 38. Two numbers are in the ratio of 3:5, but if 10 be taken from the greater and added to the smaller, the ratio is reversed. Find the numbers. 39. Two bodies are moving at uniform rates. The first goes ra feet in a seconds and the second n yards in b minutes. What is the ratio of their rates ? 179. Proportion. A proportion Is the statement of the equaUty of two ratios. Thus, 3 : 4=15 : 20, since | = i^. Therefore, 3 : 4=15 : 20 is a proportion, or 3, 4, 15, 20 are said to be in proportion, or they are said to be proportionals. RATIO AND PROPORTION 257 If a, b, c, d are in proportion or a ■.b=c : d, , a c then r = J ' a ad=bc. In the proportion a:b=c:d, a and d are called the extremes and b and c the means. Since ad^bc, it is seen that <^e product of the extremes is equal to the product of the means. 180. Fourth Proportional. When a:b=c:d, d is called the fourth proportional to a, b, c. Thus, if the fourth proportional to 10, 12, 15 is x, 10 15 then 10 : 12=15 :a; or yg = — , .-. 10x= 12x15, x=18. 181. To find a Ratio, by Solving an Equation. From certain types of equations in x and y, the value of the ratio oi x :y may be found. Ex. 1. — If bx=Qy, find the ratio oi x .y. Since 5x = %, .*. X = ft/, X _ 6 the ratio of x : y = | or 6 : 5. Ex. 2.— If 3a;-|-4t/=3t/-7x, find - 3x+7x = Sy — iy, ■ y~ 10" If each term in the equation is of the first degree in x or y, the ratio ot x : y can be found, but it can not be found if there is a term not containing x ox y. Thus, from 2x — 7i/= 10, the value ol x : y can not be found. S 258 ALGEBRA Ex. 3.— If 2x^—lxy-\-6y^=0, find x:y. Factoring, {x-2y){2x-3y) = 0, :. x—2y = 0or2x—3y = 0, - = 2 or 5 • y 2 Here there are two values of -. If we divide each term of the given equation by y*, we get In this form we see that the equation is a quadratic in - , and we might naturally expect to find two values for the required ratio. Ex. 4.— It 2x—5y-\- 2=0, 3x+2«/— 2z=0, find the ratios of x, y, z. If we eliminate z in the usual way, we get 7x-8y=0, x _8 X 8 ~ y i' If we eliminate y we get X 8 2 "" 19 X 8 ~ z l9" We can combine these results in the convenient form X 8 ~ y 7 2 ~ l9 EXERCISE 128 (1-21, Oral) Find the value of x in the proportions : 1. 2 4 3 x' 2. 3 9 7 x' 3. 2 X 5~3' 4. x_5 6~4" 5, 3 6 X -12 6. 5 7. a c h~x' 8. a X b~~c' 9. 4_ X a:~I6' 0. 3:7 = 12: a;. 11. 2:3=a:: 9. 12. x: 5 = 7: 10. RATIO AND PROPORTION 259 Find the value oi x: y, 13. 2x=ly 14, 32/= 12a;. 15. 2x-i/=0. 16. \x=^'iy. 17. 2x=-Zy. 18. S^z+llx^O. 19. a:2=42/2. 20. 4x2=91/2. 21. (x-Sj/Kz— 5i/) = 22. If - = -, show that - = -, and - = -. b d c a a c (^ Find a fourth proportional to : 2, 3, 18; 5, —7, —10; k, \. \ ; a, b, c ; a, 26, 3c. ^4y Find a fourth proportional to: a — h, {a-\-bY, a^—b^; and -3a+2, a2— 5a+6, a2-5a+4. What number must be added to each of the numbers 2, 4, 17, 25 8(^Jfiat the results will be proportionals ? (Verify.) (2&7 If a+x, 6+x, c+x, d-j-^ are proportionals, find x. What does the result mean when bc=ad ? 27. Find a in order that a+3 : a+15=3 : 4. 28. ^'s age is to B's as 4 : 5. Five years ago the ratio was 3 : 4. Find their, ages. 29.* In an equilateral triangle the ratio of the altitude to either of the equal sides is VS : 2. If the altitude is 10 inches, find the side to two decimal places. 30. When a line is drawn j^arallel to the base of a triangle it divides the sides in the same ratio. In the figure, AB=20, AD—l'i and AC=15. Find AE and EC. 31. In the figure, the triangles ADE and ABC are ^ similar. When triangles are similar their correspond- / \ ing sides are in the same ratio, so that D/. \E AD : AB=DE : BC=AE : AC. ^ ^ If AB=8, BC^IO, AC=9, AD=6, find the lengths of all the other lines in the figure. 32. In the same figure, the areas of the similar triangles are in the same ratio as the areas of the squares on their corresponding sides. If AD^20 and AB=35, find the area of ADE if the area of ABC is 735. s 2 260 ALGEBRA 3 D 9 33 The side of the square A BCD is 10 inches and EF is parallel /^ B to DC. If the length of AE is 3 inches, find the length of FC to three decimals. 34. If the bases of two triangles are in the ratio 3 : 4 and their heights in ratio 8 : 9, find the ratio of their areas. J3tf^ From these equations find z : y, 13x+5!/=9a;-f-13j/ ; ax+by=cx+dy ; mx—ny^=nx-\-my ; px-\-qy=0. as^ Find tv.'o values ol x : y when 6x^—l3xy+6y^=0 ; x^=4xy+5y^. ,>?r^ If 5a — 36+2c=0 and a + b+c=0, find the ratios oi a : b, a : c, a : b : c. 38. Find the lengths of all the other lines in this figure. 39. If a pole 10 feet high casts a shadow 17J feet long, what will be the length of the shadow cast, at the same time, by a monument 84 feet high ? Write the equation 3x^-'\0xy-{-3y^=0 in a form showing - as the unknown, and find x : y. J^. A number of two digits bears the ratio 7 : 4 to the number formed by reversing the digits. If the sum of the numbers is 66. find them. ^2. The length of a room is to the width as 6:5, and the length i/ to the height as 3 : 2. If the area of the floor is 187^ square feet, find the dimensions. 43. If 4 men and 3 women earn as much as 16 boys, and 6 men and 5 boys earn as much as 10 women, find the ratio of the earnings of a man, woman and boy. 44. If 3ab+2b^:2a^-ab=9:5, find a : b. 46. When the angle A is bisected, AB : AC^BD : DC. (1) If AB=10, AC=S, BC=12, find BD and DC. (2) If AB=c, AC=h, BC=a, find BD and DC RATIO AND PROPORTION 261 46. The ratio of the area of a rectangle to the area of the square described on its diagonal is 6 : 13. Find the ratio of the sides. 47. The sides of a triangle are 7, 10 and 12. The perimeter of a similar triangle is 72J. What are its sides ? 48. If — — = = -~ — ^-! — , find x-.y.z. 5 7 9 182. Mean Proportional. When three numbers form a proportion, it is understood that the middle number is to be repeated. The three numbers are said to be in continued proportion, and the middle one is called the mean proportional between the other two. Thus, 4, 6 and 9 are in continued proportion, since 4:6 = 6:9. Here 6 is the mean proportional between 4 and 9. If X is a mean proportional between 3 and 27, - = ~^, .. x» = 81, .-. a;=+9. X 27 ~ .'. the mean proportionals between 3 and 27 are + 9. 3 9 3—9 Since i: = tt;^ and — ;; = -— -, it is seen that these are the correct 9 27 —9 27 results. Similarly, if x is the mean proportional between a and h, then a X /— f . - = -, .-. x=-»-Va&. X Therefore, the mean 'proportional between any two quantities is the square root of their product. 183. Third Proportional. If a, h, c are in continued proportion, c is called the third proportional to a and h. Thus, if X is the third proportional to 6 and 15, -| = lf, • 6x=225, .-. x=37i. 16 X 262 ALGEBRA BXSRCISB 124 •o. Find a mean proportional between 4 and 16 ; 2a and 8a ; 4063 and 'da% ; (a-i)^ and (a + ft)^. J?r Find a third proportional to 2 and 4 ; 3 and 30 ; 5a and lOaft ; x'^—y^ and x— ?/. A. The mean proportional between two numbers whose sum is 34 is 15. Find the numbers. (/4. Three numbers are in continued proportion. The middle one is 12 and the sum of the other two is 51. Find the numbers. 5. What number must be added to each of the numbers 3, 7, 12 80 that the results will be in continued proportion ? C* In the figure, the angle BAC being in a semicircle is a right angle. When AD is drawn P perpendicular to the hypotenuse it is proven in u geometry that AD is a mean proportional between BD and DC, AB between BD and BC, and AC between CD and BC. (1) If BD= 4, DC= 9, find AD. (2) If BD= 5, AB= 8, find DC. (3) If BC = n, AC=\2, find DC, AB, AD. (4) If AB= 3, AC= 4, find BC, AD, BD. 7. How would 30U use the preceding to find (1) A line whose length is VG inches ? (2) The side of a square whose area is 12 square inches ? A. Find two numbers such that the mean proportional between tnem is 4 and the third proportional to them is 32. 9^ Divide a line 21 inches long into three parts such that the loV^est is four times the shortest and the middle one is a mean proportional between the other two. 184. The following examples will illustrate a method which has many appplications to problems with ratios or fractions. RATIO AND PROPORTION 263 ^^- ^-^^ 6 = d' P^^^^ '^^' ^^:i56^ = ^^35^ • Since r = -„ let each fraction = k. a Then r = ^. .'. a=bk, -, = k, .'. c=dk. b d Substitute these values of a and c in each side of the identity to be proven. 3o«+2b« _ Sb^k* + 2b* __ b*(3k* + 2) _ 3k* + 2 a*— 5b* ~~ b*k*-5b* ~ b^k^-5) ~ fc2-5 * 3c' + 2d» _ 3d*P + 2d * _ rf»(3fc* + 2) _ 3A:'' + 2 cs_5d» ~ d*A;«-5d« ~ d^(k^-5) ~ k'-5 ' 3a» + 2 6» _ 3c» + 2d' Ex. 2.-If - = ^ = - . prove ^^^qip^q:^ = (a+6+c)3' Let x = ak, y = bk, z = ck. Substitute as before and show that each of the fractions is equal to k^. ,. « c ^^ a-\-b c-\-d 185. If r = T . then — ^ = — !— ^ • 6 rf a—o c—d Prove this by letting a = bk and c = dk, as in the preceding examples. Here the fraction ^ was obtained by adding and sub- tracting the terms of the fraction r, and . was obtained c — a in a similar way from -^ • This principle is sometimes useful in simplifying equations. 4:X-\-3 3a+46 Ex. 1.— Solve 4a:— 3 3a— 46 Adding and subtracting, ^ = Si* .-. 646x = 36a, _ 9a ^~T6b' Solve also, in the usual way, by cross multiplication. 264 ALOE BE A „ ^. a-\-b-\-c4-d a+b—c—d a d Ex. 2.— If -—-^ — '—J = —4- ^ . prove r = - a— 0— c-f-a a—o-\-c—d o c AAA- A u, *• 2a+2d 2a-2d Adding and subtiacting, ^^-^^ = ^^—^, a-\-d a — d b + c~ 6 — c' o + d _ b-{-c a—d b—c Adding and subtracting, 9^ ~ 9~ ' a b d c * a _ d •'■ b^c' EXBRCISB ;2$ ^. If - = -, prove that — = -—% •^ h d^ 2a+36 2c+3d ^. If a : 6=c : d, show that ma+nfe : ma—nh=^mc-\-nd : mc — nd. 3< If a : 6=c : d, prove a26d+6^c+6c=a62c+a6d+od. • If ? = ?, find the value of ??!±^. •^ y 4 6x2/4-2/2 (Here x=|?/, substitute for x and simplify.) T* a; y ^ , ,, , , (2x+3w)(3x+2w) 5. If - = ^. find the value of \^ /,,„ /, • 2 3 (5x— 32/)(3x— 5t/) ^ If ? = 3 and ? = ?, find the value of "P^ . ^-^ y 6 5 2az+3by fii If - = - = -, prove that each fraction is equal to — ^ — , that ^^ a b c o+fi-t-c . , sum of numerators IB, to sum of denominators £\ n -^ = J^= -^, prove x(6+c)+2/(c+a)+2(o+6)=0. \J b—c c—a a—b RATIO AND PROPORTION 265 ^? If .1, b, c, d are proportionals, prove that ah-\-ac^, h^d-\-h(J^. d-c^ and h-d"^ are proportionals. b c , prove that a^h-\-c. +y y—z z+x (3- n- ^"^ X \^. If o : b=b : c, show that a : c=a^ : b^. /i2J If a—b:a-\-b^=c—d:c-\-d, prove a:b=c:d. 6 T, a+6+c+d a — 6+c — d , -1^0 c If , — ! — = — ! ; , show that - = - . a-\-b—c—d a — b—c-\-a b d c , 3x+46 5a+3b ooJve = — - • 3x-46 5a- 36 16. If the sum of two numbers is to their difference as 7 to 4, find the ratio of the numbers. ^a T* a—b+l b—c + 3 c—a+5 , ,, , . x i- 16. If ; = • = — , show that each fraction b—c+2 c—a+4: a— 6+6 equals f. (Use Ex. 7.) 17. Solve ^^±^=^^±^±^. aa;— 6+c bx-\-c—a ^. If a : 6=3 : 5, 6 : c=7 : 9, c : d=^\f>-, 16, find the ratio oin:A. s«i>. 19, If - = ^ = - , show that each fraction equals — ■ ^— I — , and a b c ^ 5a— 36+2c also equals j — — • ma-\-nb — pc y^^. Find two numbers such that their sum, difference and product are proportional to 4, 2, 9. 21. If a, b, c are consecutive numbers and if c^—b^ : 6^— a^=41 : 39, find the numbers. 22. The length and breadth of a room are as 3 : 2, and if 2 feet be added to each, the new area of the floor is to the old as 35 : 27. Find the dimensions. ^^23. If a : b=c : d, prove a : a+6=a+c : 0+6+c+d. A ^ '4. If -^^^^ show that ? = -^ lOc+d 12c+d 6 d If a : 6=6 : c, then 0^+06 : 62=62+6c : c' /f, 266 ALGEBRA BXBRCISB 126 (Review of Chapter XZ) Write as fractions in their simplest forma : w^7i:8|. ^. x*-y*:{x-y)\ ^. o« + 6» : (a + 6)« 4. l--i:l + -. 5, a* :1 6^ a-- : c ^_ 1 1 „ ^ . , 1 ,1 >c^a»+l + ^:a-l+^ ^ ■ x*-5x+6' x'+z—12 x^ i 9. Divide 144 into three parts proportional to 3, 4, 11. * *^ 10. What must be added to each term of 4 : 7 to make it equal /* 4 to 6 : 7 ? r 11. Write as a proportion in two ways : 3. 6 = 2. 9; 2.5 = 3z; ab = cd ; (a + 6)(a-6) = 3 . 4 ; a*-5o + 6 = a* + 5o+4. ^i^ If the means are 7 and 12 and one extreme is 3, what is the :)ther extreme ? 131*. Find a fourth proportional to: 7, 15, 35 ; 11 a, a', a' ; x + y, x — y, x* — w* ; 7. — — r , a* — b*. a — a-}-o IfC Find two numbers in the ratio 9 : 5, the difference «)f whose aquare^s 504. IS. Two numbers are in the ratio of 5 : 8, and if 8 be added to the 1^3 and 2 be taken from the greater, the ratio is 14 : 15. Find the numbers. i^. Find two numbers in the ratio 6 : 5 so that their sum is to the difference of their squares as 1:3. 17.* If the ratio a — x : 6 — x is equal to the square of the ratio a : h, find X. ^a. If 2x+3?/ : 3x-5i/ = 9 : 11, find x : y. WT If (5x-72/)(2x-32/) = (4x-52/)(x-j/), find x : y. 20. If 4x-52/ = 2x+2?/, find %x+2y : 2x+32/. 21. If 6x«+152/*=19x2/, find X : 2/. ^ Ifx* + x+l : 62(x+l) = x*-x+l : 63(x-l), find x. 23. If 2x + 2/-23 = and 7x+6z/-92 = 0, find x:y,x:z and x-.y.z 24. If - = s. find the value of '^^^- • y 3 3x+llj/ X RATIO AND PROPORTION 267 Find a mean proportional to a;* j and y^ ^ • y ^ 26. If az + by : bx-\-ay=9 : 11 and a : 6 = 3 : 2, find the ratio of X to y. 27; If r = J = 7' show that each of these fractions is equal to ma—nc—pe mb—nd—pf j<? 28. Find two numbers whose sum, difference and product are proportional to 5, 3, 16. a+b 2o*-36« 2c--3d» X ^ If a •.b = c : d, show that c+d' 2a* + 36* 2c*+3d»' {ab + cd)* = (a* + c*)(b* + d*). u«^ If r- = — 7 = —-T , prove that o-\-c—a c-\-a—o a-\-b—c x(b — c) -\-y{c — a) -\- z(a — b) = 0. 31. If any number of ratios are equal, show that each ratio is equal to the ratio of the sum of all the antecedents to the sum of all the consequents. 32. If 3x-22/ + 42 = 2a;-32/ + 2 = 0, find the ratios of x, y, z. If also, a;* + 2/* + 2*= 150, find the values of x, y and z. ^[3'r The hypotenuse of a right-angled triangle is to the shortest iide as 13 : 5. If the perimeter is 120, find the sides. 34. The length, width and height of a room are proportional to 4, 3, 2. If each dimension be increased 2 feet, the area of the four walls will be increased in the ratio of 10 to 7. Find the dimensions of the room. d C 6 CL •\— C^ — l-C CLC& 35. If r = - = -, show that ,-r rr — r- = 5-r; • b d f b^ + d^+P bdf 36. If the sides of a triangle are 6 and 8 and the base is 4|, find the segments of the base when the bisector ot the vertical angle is drawn. 37. If yiZ^ = ^^P^, show that z* = x*+yK z-x+y x-\-y+z' ^ 38. The incomes of A and B are as 2 : 3 and their expenses are Bf 6:7. If ^ saves 25% of his income, what % does B save ? 39. Find three values of the ratio x : y ii 3{x*-ixh/ + 5xy*-2y*) = 2{x'-2xh/-2xy* + 3y»). CHAPTER XXI THE GENERAL QUADRATIC EQUATION 186. Type of the General Quadratic, The equation is called the general quadratic equation, because every quadratic equation may be reduced to this form. If the factors of ax^-\-bz-\-c can be obtained, the roots of the equation can be found by solving the two equivalent equations. 187. Solution of Literal Quadratics. The methoa of completing the square may be applied to the solution of quadratic equations with literal coefficients. Ex. 1. — Solve z^-}'2mx=n. Complete the square by adding m' to each side, x* + 2mx-{-m* = n + m*. Take the square root, x+Tn= ± Vn+m*, x= —m+ Vn-f-w*. The two roots are —m+Vn+vi*, —m—Vn-^m*. Ex. 2.— Solve z^-\-pz-\-q^O. Transpose the absolute term, x*+px= —q. p* p* p* p* — 4o Add ~- to each side, x* + px + ^ — — 9 + ir — — ^ — ' 4 4 4 4 Take the square root, x + 2=± 2 ' -p Vp*-4q X = -t:— ± ;; THE GENERAL QUADRATIC EQUATION Ex. 3.— Solve aa;2+6a;+c=0. Divide by a to make the first term a square. 26& Transpose, r* + -r + - = 0. a 'a X* -\ — X — — Add -r—. to each, x* H — x -\- -r—z = fe*— 4ac 4a» Take the square root, 4a« ^ + 2^=± 4a* a 4a^ v'62_4ac 2a h . V6*— 4ac .'. aJ = — K- + ;j ' 2a - 2a _ -6±\/&2— 4ac ~ 2a The roots of the general quadratic equation are — &± V&^— 4ctc 2a 188. The roots of the general quadratic might also be bund by factoring as in art. 171. ax*-[-hx-\-c = 0, \ a a] (f , by^c 6M Since the product is zero, one of the a is not zero, as the equation would not then ^ + 2^ + must be zero. But quadratic. 270 ALGEBRA eixbrcise: 127 Solve by either of the preceding methods : 1.* a:»-2aa:=3a2. 2. a;2+46z-56*=0. 3. z*— 6mx-H3m2=0. 4. x^+4:px—p^=(i. 6. z*-2az+6=0. 6. x^^2bx-c=0. 7. o«»+2ax=6. 8. ax2-i-26x+c=0. 9. az*— 6z— c=0. 10. px'^—qx+r=(). 189. Solving by Formula. The roots of any particular quadratic equation may be found b}" substituting the values of a, h and c in the roots of the general quadratic. Ex. 1.— Solve 6a;2-7a;+2=0. Herea = 6, 6=-7, c = 2. „ , . , , . — 6+V6>-4ac Substitute these values in x= ^^ » 2a + 7±V49-48 •■• ^= 12 7±1 8 6 2 1 = 12 =r2°'r2 = 3°'2- Verify by substitution. Ex. 2.— Solve 5a:2+6x-l=0. Here a=5, 6 = 6, c= — 1. _ -6±\'36-(-20) _ -6+^56 •'• ^~ 10 ~ 10 ' -6±2Vl4 -3±VIi ~ 10 ~ 5 ■ In this case the roots are irrational, but, if necessary, we may substitute for Vli its approximate value 3-742, when the roots become -3±3-742 -742 -6-742 ,_ , _. or z. = -HS or — 1-348. 5 5 5 Note. — The pupil ia warned to be careful of the signs whoD Rubatituting, particularly when c is negative. THE GENERAL QUADRATIC EQUATION 271 Ex. 3.— Solve 2a;2— 5x+6=0. 0=2, 6=— 6, c = 6. + 5+V'25-48 5+V^=^ ••• * = -"^-4 = ^4 190. Imaginary Roots. In the preceding result the numeri- cal value of the roots cannot be found even approximately, for there is no number whose square is negative. Such a quantity as V— 23 is called an imaginary quantity, and the roots in this case are said to be imaginary. This is merely another way of sajdng that there is no real number which will satisfy the equation 2x'^—6x-\-Q=0. 191. Methods of Solving Quadratic Equations. When a quadratic equation has been reduced to the standard form, it may be solved : (1) By factoring, by inspection or by completing the square. (2) By substitution in the general formula. The pupil is advised to try to factor by inspection,^ and if this method is unsuccessful, then substitute in the general formula. As the general formula will be used very frequently, it is absolutely essential that it be committed to memory. The roots of ax^+ox-\-c=0 are 2a EXSRCISB 128 Solve, using the formula : 1.* 3x2-5x4-2=0. 2. 24x2-46x+21=0. ^^. 575x2-2x=l. 4. 2x2-6x-l=0. ^O 5. 247x2+5x=12. X 2x2- 13x+ 10=0. 1 JJff 391x2+4x=35. l^ 1200x2- 10x=l. ^ ■^ ft. x^+x{3b-2a)=6ab. i>a. 2x2-25x+77-^0. }y 6x^-x-l=0. ^ 1800x«-5x-I=0 272 ALGEBRA Solve by any method. Verify 13-18 : 13. 27a;2_24a:=16. 14. 15x2+7a:-2=0. 16. 4a;2-17a;+4=0. 17. 460x2-3x=l. 19. 9x+4=5a;2. 20. 3a;2+2=9a;. 22. 25. 28. 30. 9x+4=5a;2. 4x2-4a;=79. 23. f-2/=2/2 26, X - -^ =0. a;+2 (x-4)2-3(x-9) = 15. 2ax2+x(a— 2)=1. 32. 2x(x-2)=a2-2. ^4-1 = 2 ^2 29. 31. 33. 15. 18. 21. 24. 27. 12x2-T-6=0 5-26a;+5x2=0 2x2-2x=§. 1 * 2 3"'"9'"x" 2 7 , w (x-2)(x+3)=x(5x-9)-2. ac+ - = - + ex. a X X 2 2-^x ^36. ^7. 38. x+1 ^x+2 x'^—xy—^y^- 12 x+3' -12. -^5. (x+2)2 n y=2, find X. X 3 3 + x' -(x + 3)2=(x+6)*. n £ x+1 „ , , ., , , x+1 2x Find the sum of the roots of x^ , find X to three decimal places. 3x=20. 39. 40. The area of a square in square feet and its perimeter in inches are expressed by the same number. Find the side of the square. 41. The length of a rectangular field exceeds the width by 16 rods and the area is 32 acres. Find the length. 42. Find three consecutive even numbers whose sum is | of the product of the first two. 43. A line 10 inches long is divided into two segments so that the square on the longer segment is equal to the rectangle contained by the whole line and the shorter segment. Find the segments to two decimal places. 44. Find two numbers whose difference is 3 and the sum of whose squares is 317. 45. The area of a square is doubled by adding 5 inches to one side and 12 inches to the other. Find the side of the square. TOJD QiUNERAL QUADRATIC EQUATION 27S 46. Three times the square of a number exceeds eight times the number by unity. Find the number to three decimals. 47 Mr. Gladstone was born in the year a.d. 1809. In the year A.D r^ he was x— 3 years old. Find x. 48. The area of a rectangular field is half an acre. The perimetei is 201 yards. Find the sides. 49. One root of x^—bx-\-d=(} is 8. Find the value of d and the other root. 50. If a train travels 10 miles per hour faster than its usual rate, it will cover 480 miles in 4 hours less time. Find its usual rate. 51. Divide 3 into two parts so that the sum of their squares may be 4§. 52. I buy a number of articles for $4'80 and sell for $5'95 all of them but 2 at 6 cents a dozen more than they cost. How many did I buy? 53. A straight line AB, 12 inches in length, is divided at C so as to satisfy one of the following conditions. Find, in each case, the length of 4 C to two decimals : (1) AC^=2BC\ (2) AC^=2AB . BO. (3) 3^C2=44£ . BG. (4) AC^+^BC^=2AB\ (5) AC^-BC^^IQ s [. in. (6) AC{AB^BC)=2 sq. ft. 54. I buy a number of books for $6, the price being uniform. li they had been Subject to a discount of 5 cents each, I could have bought 6 more for the same money. What did each cost ? 55. Solve the equation ax^-\-bx-\-c=^0 by multiplying by 4a and completing the square of 2ax-\-h. =^ c 1 3+2a; 2— 3a; , \^x-x^ 1 56. Solve — ■ = - . 2-x 2+a; a;2-4 3 Verify the roots obtained. 192. Equations Solved like Quadratics. There are certain types of equations of a higher degree than the second, which may be solved by reducing them to the form of quadratics. T »W4 ALGEBhA Ex. 1.— Solve x*- 10x2+9=0. This is ail equation of the fourth degree, but we might write it ii the form of a quadratic, thus (x*)*-10(2:«) + 9 = 0, or if we write y for x* it takes the form j/»-102/ + 9 = 0, .-. (3/-9){t/-l) = 0, 2/ = 9 or 1. But i/=aj*, .". x* = 9 or 1, a;=±3 or ± 1.- We see that this equation has four roots. This is what we might expect, as it is an equation of the fourth degree in x. Verify each of the four roots. Ex. 2.— Solve (x-2_5x)2+4(x2-5a;)-12=0. Here we consider x* — 5x as the unknown, whose value should first be found. Let x»-5x = =y> .-. y ' + 4y- 12 = =0, :. {y ^6)(t/~ 2) = =0, y = .- 6 or 2. x* — 5x= —6, or x»-5« = 2. x*-5x + 6 = 0, 2. ; x«-5x-2 = 0. (x-3)(x-2) = 0, 2; = 3 or 6±V254-« 5-fV33 This equation hsis four roote, two of which are rational and thf= Dther two irrational. Verify the rational roots. Ex. 3.— Solve (2a;2-|-3a;-l)(2x2+3x-2)=-56. Let 2x*-^Zx = y. m.- ,. 3 , -3±V^^9 The result la x = ^ , — 3. * 4 THE GENERAL QUADRATIC EQUATION 271 w^ A. c 1 -^'4- 2a; ^ 3 26 bX. 4.-Solve _^+^,^^ = -. 1 26 Complete Ex. 's 3 and 4 and verify the rational roots. Ex. 5.— Solve z3-1^0. Factoring, (x— l)(x*-f-x+ 1) = 0, .-. X— 1 = or x* + x+l = 0, -1±V^ x= 1 or X = ^ • We thus see that if one root of an equation of the third degree, or a cubic equation, can he found by factoring, the equation can he completely solved. This equation might be written x^ = l, and each of the thi-ee roots when cubed must give unity, which shows that unity has three cube roots. This is what we might have expected, as we have already' .seen that unity has two square roots. + 1 and —1. BXERCISE 128 Solve and verify the rational roots : ,^. x*-5x2+4=0. ^J.. x«- 13x2+36=0. 3. V4-12=31y«. ^ 8i«-65x3+8=0. ^. (.H5.^6)(.«-9.+ U)=0. ^ £!i^+_^ = 2. i. (x2-4x+o)(x2-4x+2) = -2. ^t. x» + a; + 1 = ^^ .,«**♦* ., a:2+x r ^-9. (*2+x4-l)«-4(x2+«+l)+3=0. ''!•. x3-4x2-4x+16=0. ,11. 6/'x + l)^-35/'x + lW50=0. (i+x+x2)(x+x»)=156. t2 lA- 876 ALGEBRA 13.* (x+l)(«4-2)(a:+3)(a;+4)=120. (Multiply the first and last factors and the second and third.) 14. a;(x-l)(x-2){x-3) = 360. 16. Find the three cube roots of 8 by solving the equation x'— 8=0. 16. Find the four fourth roots of 16 by solving the equation :fc4_16=0. 17. Solve x'— 19x4-30=0 being given that 3 is one of the roots, ^ 18. Solve 12x3— 29x2+23x— 6=0 (ygg ^i^g factor theorem). 19. It is evident that 4 is a root of the equation x(x-l)(x-2)=4.3.2. Find the other two roots. 20. Find the six roots of 8x« — 217x3+27=0. 21. Solve(x2— x)2— 8(x2-x) + 12=0. 22. Solve x2 + jL + a; + 1 = 4 (^Add to x? + i- the quantity X^ X \ x^ required to make it the square of x H ) X ) BXERCISB 180 (Review of Chapter XZI) ^\. Explain the different methods of solving quadratic equations Illustrate them, by solving in full the equation 3x* — 4x— 15 = 0, by each method. ^. Solve 323x* + 2x=l. (^ The difference of two numbers is 8 and the sum of their squares is 104. Find the numbers. V/C If X = 2( 1 + - j, find. X to two decimal pie 5. What is the price of meat per lb. if a reduction of 20^ in the price would mean that 5 lb. more than before can be bought fo^ $3 ? sji. Solve 10x»-I9x-9 = 0. \^. The sides of a right-angled triangle are o, o— 10 and o+lO. What are the sides ? j/ioT THE GENERAL QUADRATIC EQUATION 27'3 x—\ x — 5 Solve H = 4. X—i x—o The sum of two numbers is 45 and the sumi of their reciprocals 9. Find the numbers. Solve 6375x»-10a;=l. 11. The length of a rectangular field is 5 rods more than the width. The area is 3 J acres. Find the sides. 12. What miist be the values of n in order that —: r—— may 10n+21a ^ equal | when a = ^\ ? \13. The perimeter of a rectangle is 56 and the area is 192. Find the diameter of the circle which passes through its angular points. 14. Solve OOTox'+Tox^ 150. 15. By solving (x— 2){x— 3) = (a— 2)(a — 3), find a quantity whicL can be substituted for a in (a — 2)(a— 3) without changing its value. ,^. Solve x» - 2a; » - 89a; + 90 = 0. 17. Two trains each run 330 miles. One of them, whose average speed is 5 miles per hour greater than the other, takes ^ an hour less to travel the distance. Find their average speeds. « 1 a;»+2 , x+l ,» Solve ^:j-j-+^-j:p2 = 2*--^ V. o , a; , x*+\ 109 9. Solve ^^^+-^ = ^. 20. I sell a horse for $96 and gain as much % as the horse cost in dollars. WTiat was the cost ? ^. Solve {a;*-3x-5)* + 8(x*-3a;-5) + 7 = 0. 22. Divide 25 into two parts so that the sum of the fractions formed by dividing each part by the other may be 4 25. 23. The sides of a rectangvdar field are x+17 and a;— 17. The diagonal is 50. Find the area. 72 Solve x» + a;+-r-— =18. x'-{-x Solve (m» — n»)x» + 2x(m* + n*) + m* — n» = 0. Solve (x-5j(x-J )(x+V)tx+ 3) = 60. 27. Find all the roots of the equation x'= 1 25. 28. Since x* — 8x+ 12 — (x— 2){x — 6), for what values of x will the expression x* — 8x+12 be equal to zero, and for what values will it be negative ? 278 ALGEBRA 29. Solve -. = ~ ■ x — a x — b a-\-b 30. Solve adx — OCX* = hex — hd . 31. The area of a square is trebled by adding 10 inches to one side and 12 inches to the other. Find the side of the square. 32. Solve x«(a«-c*)-a;(o6 + 36c) -262 = 0. 33. Solve (x* + 6x+8)= + 3x(x2 + 6a;-(-8) = 0. 34. A man bought a number of acres for S300. If he had paid S5 more per acre, the number of acres would have been 2 less. Find the number bought. 35. Solve / , , = - + i + ^ • x-\-a-j-o X a b o« « 1 x — a , x~b b a 36. Solve —z — = 1 = ' b a x — a x — b 37. OX and OY are two roads at right angles. A starts at noon along OX at 3 miles per hour. B starts at 2 o'clock along y at 4 miles per hour. Find to the nearest minute when they will be 20 miles apart. 38. Solve a*x»-2o»a; + a*-l = 0. 39. Solve ax' , =cx — 6x*. + 6 40. A gra%'el path 2 yards wide is made round a square field and it is found that it takes up j^^ of the area of the field. Find the area of the field in square yards. 41. Solve 8 = vt+\(>t* ioT t. 42. What positive integer is that, the sura of whose square and cube is nine times the next higher integer ? 43. Solve (x« + x-2)«-4(x* + x-2) + 3 = 0. 44. The side of a square is 34 inches. Find at what points in the sides the vertices of an inscribed square must be placed so that it may have an area of 676 square inches. 45. Write the equation ox* + 6x2/ + C2/*=0 aa a quadratic in - . What are the values of - and of - ? y X 46. What positive integral value of x will make x*+10x moat nearly equal to 1000 ? CHAPTER XXII SIMULTANEOUS QUADRATICS 193. Consider the problem : The sum of two numbers is 12 and the sum of their squares is 74. Find the numbers. Let a; = one of the numbers, 12 — a: = the other, .'. a;* + (12-x)« = 74. Solve this equation and find x=7 or 5. If x = 7 or 5, then 12 — x = 5 or 7. .*. the numbers are 5 and 7. Here we have used only one unknown. We might have solved by using two unknowns. Let X and y be the numbers, .-. a: + t/=12, and a;* + 2/* = 74. How can we obtain from these two equations the original equation in the preceding solution ? 194. Type I. Ex. 1.— Solve x+Sy=lO, ih x^-\-xy=4. (2) From (1), x=l0-3y. (3) Substitute in (2), (10-32/)* + 2/(10-32/) = 4, •*• 100 - 60y + 9y^+lOy- 3yt = i, Qy^-50y+96 = 0, 32/*-25i/ + 48 = 0, (y-3)(32/-16) = 0. .*« 3/ = 3ori^. 370 280 AL0E3RA Substitute y= 3 in (3) and x=l, .. y=V .. ,. ,, x=- -6. There are therefore two solutions, x=l, j/ = 3 ora;=— 6, y = 5f x=l or -6. j/ = 3 or 5 J. Verify by showing that x=l, 2/=» 3 satisfies both equations and also x= — 6, 2/ = 5i. The pupil must note that a;=l was obtained from i/=3, not from «/=5^. Therefore, x=l, y = ^\ is not a solution, nor is a:=— 6, y=3. Verify this by substitution. Equation (1) is a linear equation, or an equation of the first degree in x and y. Equation (2) is a quadratic equation, or an equation of the second degree in x and y. A sj'stera of equations of this type, that is, where one is of the first degree and the other of the second degree, may alwa3^s be solved by the method of substitution, which does not differ from the similar method employed in art. 107, when both equations were of the first degree. Ex. 2.--Solve 3x- y=5, (1) a;24-3a:y=15. (2) From (I), i/ = 3x-5, .-. x* + 3a;(3x-5) = 15, .-. 10x*-15x-15 = 0, 2x*-3x-3 = 0, 3+V33 3±5-74o 4 4 = 2186 or --686. • j/ = 3x-5= ^^-/^^^ =1-558 or -7058. J 4 Here the roots are irrational and it is customary to leave them in that form, unless the decimal form is asked for. SIM ULTANSO US Q U ADR AT W 8 BXERCISE] 181 Solve and verify 1 -6 : 1 x^y^l, 2/ x-y='k, /3. x-2y=Q, xy=12. xy=60. x^-y^=27. ji. x-y=S, ^ x-y=6, 0. 2x+2/=9, a;2+y2=65. a;2-2/2=60. x^-y'^=\b. 1* x+Zy=U, 8/; 2x+Sy=l2, 9^ 3a;-4t/=2, ^ x2+2/=27. V a:2+2/2^13. '^ 3x^+2y^=U0- \^tO ) x2+3a»/ '-«/2_^2x=3'' Kll\ 3:K2_2a;2/+5x-?/=17 X— j/=3. I J 2x -3i/=-] 12. If x—3y=2 and x2— a;?/+2?/2=6, tind the valu8»» of x and i/ to three decimal places. 13. The hypotenuse of a right-angled triangle is 25 and the perimeter is 56. Find the sides. 14. .4 is 10 years older than B. Eight years ago the sum of the squares of the numbers representing their ages was 148. Find their 15. The diagonal of a rectangle is 50. The difference of the sides is 10. Find the area. 16. The area of a right-angled triangle is 96 and the difference of the two sides about the right angle is 4. Find the hypotenuse. 17. Solve 3x+5y=2, 3x^-l0y^-xy+28=0. 18. K each digit of a number be increased by 2, the product of these increased digits will be the original number. When the digits are interchanged the resulting number is thirteen times the tens digit of the original number. Find the number. 19. The sum of the areas of two squares is 40 square inches. The side of the smaller is 10 inches less than three times the side of the larger. Find their sides to three decimals. 20. Solve —+5^= 14, 2/-l=a;. y^ y 195. When both equations are of the second degree in x and y, they can not always be solved by elementary methods. 282 ALGEBRA There are special cases in which they can be solved without difficulty. ^^,y • ' / ^ ^\ ^^ ^\J(j\JL cUc^s*^ ^^' • \s<^ . o:. r^-Cf"^ ^ ^ 196. Type II. Solve z^—5xy-\-4:y'^=0, Factoring (1), (x-4y)(x-y) = 0, x=-ly or x = y. We are now required to solve : x» + 2/» + 3x=29\ ^^^ x*+y* + Zx = 2%\ x=iyf x = y / Substituting the value of x, 16y» + 3/*+12j/ = 29, 17i/»+12j/-29 = 0, (j/-l)(17r/+29) = 0, y=\ or -H. x=4 or — Vt'- (1) (2) 22/* + 3t/-29 = 0, 3+V24I y = -3+V241 Here there are four solutions : X = 4 or — -3±V241 4 3+^241 1/ = 1 or — J^ or In this type the first equation contains only terms of the second degree. When that is the case the left-hand member may be factored and each of the resulting linear equations may be combined with the second equation, thus giving two cases of Type I. BXEROISB 182 Solve and verify 1-5 x2-?/2=0, vl. x2+xi/+y2=36. 2. x2-4xy+3y2=0, x2+y2=10. , 3. 3x*-2x2/-j/*=0, ^ x+y+y*=32. > x2+2/2+2x=12, 3a!»+2xy=y*. SI MULT Alii EOU 8 QUADRATICS 283 6. 4:c''+20a;y+V=0, (6> ^ + ^=14, 2ary+l=0. x—\=y^. ^'^ x^—xy—y==\. 6a;^+4i/2=ll2;y. 9. Find four solutions of the equations {x-y){x-2)^(i, (a:+y-6)(2/ + 3)=0. 197. Type III. Homogeneous Equations. Solve x2— xi/=6, (1) y^+^xy^lO. (2) Multiply (1) by 5 and (2) by 3 and subtract, to eliminate the absolute terms, and we get bx*-UTy-Zy* = Q. (3) This equation (3) is of the same form as the first equation in Tj^pe II. Grouping (3) with (1) we proceed as before. Factoring (3), {x-'iy){t. z + y) = 0, :. a;=3t/ or -\y. Substitute x=3?/ in (1), Substitute x = -^yin(I), .-. 9i/»-37/* = 6, 5 82/ *+j2/» = 6. y»=i. ,'. y* = 25. y=±\. .'. 2/=±5, x=±3. x= + l. Hence the four solutions are : x = 3, x=-3, 'or .or y=\. y=-\. x=-l, y = 5. or x=l, !/=-5. V^erify each of these four pairs of roots. If we had grouped (3) with (2), the results would have been the same. Show that this is true. In this type, terms of the first degree were absent from both equations. The expression on the left in each is homo- geneous, that is, every term is of the same degree. For this rearSOQ, this is called a homogeneous system. 284 ALGEBRA The pupil should be on the look out for special methods of obtaining f'-om the given equations an equation of the first degree. Here we might have done so by simply adding the equations and taking the square root. Solve it by this method. BXHRCISB 188 Solve and verify 1-9 : ^^ (jC. 3a;2-5j/2=28, Q^. 2x^-3y^=23, C^. x^-xy+y^=2\, 3xy-iy^^S. 2xy-3y^^3. 2xy-y^=15. A.^ 2z^—3xy=U, '^p' x^+xy=66, G.' x^—xy=5^, 3i/2-a;2+l=0. x^-y^=-U. xy-y^=18. 7. x^+2xy^Z2, )i 3x^-5zy+2y^=U, )^. x--4y^^20, 2y^+xy^\6. 2x^-5xy+3y^=6. xy=\2. ^.* x^-3y^=i, J^ x^+xy+y^=7, ht 32y^=^2xy+n, x-+xy+y'^=28. 3x^—l=xy. ^^ x^+iy^=lO. 13, 2x^—dxi/+9y^=5, 14. x-+xy+y^=l, ix^-10xy+l\y^=3o. 2x^-{-3xy+i:y^^2A. 15. 3x^-3xy+2y^=2x, 2x- + 3y^-xy=-ix. 10, Find, to two decimals, the real values of x and y which satisfy x^—xy=20 ana 3xy—y"=50. 17. When a number is multiplied by the digit on the left the product is 105 ; when the sum of the digits is multiplied by the digit on the right the product is 40. Find the number. 198. Special Methods. Since {x-\-y)^={x—y)^-{-'ixy, it follows that if the values of any two of the quantities x-\-y, x—y and xy are given, the remaining one can be found. Ex. 1,— Solve x+y=U, (1) xy=\%. (2) Squaring (1), x* + 2xy-\-y^=\.2l. Ftoxa. (2), 4an/ = 72. Subtracting, x* — 2xy-\-y* = A9, :. x—y=±l. SIMULTANEOUS QUADRATICS 285 If x + 2/=ll, If x+y^n. and x — y = T. and x-y=-l. a; = 9. y = 2. x=2,y=9. Hence there are two solutions : x=9 or 2, y = 2 or 9. Ex. 2. — Solve x—y=l\, xy=QO. Find {x-\-y)* by adding 4xi/ to (x — !/)* and complete the solution. Ex. 3.— Solve a;3+i/3=35, (1) x+y=b. (2) Dividing (1) by (2), x*-xy-ty^ = l, (3) Squaring (2), x* + 2xy + y* = 25. Subtracting, 3an/=18, a:i/ = 6. (4) Subtracting (4) from (3), x* — 2xy-\-y*=\, a; — 2/= + l. Complete the solution as before. Also solve by substituting x=5 — y from (2) in (1). Ex. 4.— Solve x*-\-xh/^-\-y^=9l, (1) x^+xy+y^=13. (2) x* + x*y*+y* = {x* + xy + y*)(x* — xy+y*). Dividing (1) by (2), x*-xy + y* = 7. (3) Subtracting (3) from (2), 2xy = Q, xt/ = 3. (4) Adding (2) and (4) x* + 2xy + y*=l6, .-. x+y=±4:. (6) Similarly from (3) and (4), x-y=±2. (6) (5) and (6) can be grouped in four ways, thixs : \ If 'or ^ or •^ or ^ X— 1/ = 2. x—y— — 2. x — y= — 2. x—y = 2. From these four solutions are obtained . x=3, —3, 1, —1, x= + 3or+l, or — — y=l, -1, 3, -3. J/=±l or ±3. k? 38< ALGEBRA Ex. 5.— Solve (a;+?/)2— 5(x4-i/) — 6=0, Factoring (1), Now solve (x + 3/-6)(x-t2/+l) = 0, x-f^ = 6 or — 1. x + y=6, , x + v= — It * and ' " ' xy = o. xj/ = 8. (1) (2) EXE3RCISE 184 Solve, by finding x+y and x—y, and verify 'J- J- x—y=4, xy=12. {x-y)^=l, xy=30. 8 J". x+y=S, xy= 1;">. 4. x2+w2=61 7. x'^+xy+y^=19, x+y=5. 9. 5x2 + xy+ 52/2=23, i x+y=l. ^. x^-lxy+y^=-lOl, xy=^30. x^-y^=l9, x—y^l. x^—xy+y^^Z9, x^+y^=35l. 17. x«+x2j/2 + 2/<=133. x^~xy+y^=lt {x+y)^-S{x+y)-28=0, x-y^3. (x-i/)2-»(x-2/) + 12=0, X2/=12. J^ x2+2/2=25, X — j/=l. j^ x2— x2/+2/^=57. x-y^8. x^—xy+y^=19, x+y=U. 10, x2+?/2=:89, xy—iO. p. 2x2+3x2/+2y2=.8, X2/= — 6. l4. x3+2/'=1064, ^ x+t/=.14. 1©^ X* + x2?/2 + y4 = 21, V-' x2+xy+t/2=7. 18. X*-x2i/2 + y4^13^ xy=2. x22/2- 27x^4- 180=0, x+y=8. 22j The perimeter of a rectangle is 34 inches and the diagonal is [3 inches. Find the sides. SIMULTANEOUS QUADRATICS 287 The diagonal of a rectangle is 25 and the area is 300. Find the sides. (24/ The sum of two numbers is 12 and the s^m of their squares is 72-5. Find the numbers. *'**t*'*****^ 7Cm» . _ The product of two numbers is 270. If each number is decreased byB the product will be 180. Find the numbers. ^^ T/^^a^- /R.O 26. The sum of two numbers is 10 and the sum of their reciprocals is A. Find the numbers. ^ 3- 27. Solve (a;-l)(i/+2)=9, 2xy=\5. 28. A and B are two squares. The area of A is 63 square inches .nore than B, and the perimeter of A is 12 inches more than B. Find the side of each. 29. Find two numbers whose product is 1 and the sum of whose reciprocals Is 2'i^. 30. Solve x3-8!/3=56, x-2y=2. 31. The sum of the two digits of a number is f of the number. The sum of the squares of the digits is 4 less than the number. Find the number. 32. The area of a rectangle is 1161 square yards, and its perimeter is 140 yards. Find the dimensions. 33. Solve 1 + 1 = -3, -i - 1 = 03. X y x^ y^ 34. The sum of a number of two digits and the number formed by reversing the digits is 121. The product of the digits is 28. Find tlie number. 35. Find the sides of a right-angled triangle whose perimeter is 24 inches and whose area is 24 square inches. 36. Prove, algebraically, that if two rectangles have equal area.s and equal perimeters, they are equal in all respects. 37. Solve x^+xy-\-y^=l-15, x^—xy+y-=5-2o. 38. What must be the dimensions of a rectangular field containing 7^ acres, if the greatest distance from any point in its boundarv to any other point is 50 rods ? 288 ALGEBRA 39. The sum of the radii of two circles is 8 inches and the sum of their areas is § of the area of a circle whose radius is 9 inches. What are their radii ? 40. What must oe the length of a rectangular field that contains a square rods and which can be enclosed by a fence b rods long. 199. Graphical Methods. What is the distance of the point P(4, 3) from the origin ? Since OP^=OM'^-\-MP\ .-. OP2^42+32=25, .-. OP =5. If any point {x^y) is the same distance from the origin that P is, then the point {x,y) must lie on a circle whose radius is 5 and whose centre is 0. But the square of the distance of the point {x,y) from the origin is x^-\-y'^, :. x2+i/2=25. It is thus seen that the equation x^-\-y^=25 represents a circle whose radius is 5 and whose centre is the origin. Similarly, x*-r2/*=16, x* + y* = 100, x* + y*=l8, represent circles with the origin as centres and whose radii respectively are 4, 10, Vl8. It is seen that it is a simple matter to draw the graph of the equation of the circle in the form x^-'ry^—r^. All we require to do is to describe with the compasses a circle whose jentre is the origin and whose radius is r. When the radius is a surd as in x'^-\-y^=l8, it is simpler to find a pair of values of x and y which satisfy the equation. Here a: =3, y=3 satisfies the equation, and the circle is then described through the point (3, 3). SIMULTANEOUS QUADRATICS 289 y '~ 1 ^ N j^y / \ / / X -£^- / X ^ / / 1 / / - 1_ \ / / o,'i7^ \y |_v. ^\ y L 1 200. Graphical Solution of Simultaneous Equations. Solve x^-\-y^=25, (1) x-y=l. (2) (1) represents a circle whose radius is 5. (2) represents a straight line, two points on which are (1, 0) and (0, —1). The graphs of (P and (2) are shown in the diagram. The lintj cuts the circle at the points (4, 3) and (-3, -4). .*. the roots of the given equations are (4,3) and (-3,-4). 201. Equal and Imaginary Roots. Solve, (1) a;2+i/2^18, x-y=0. (2) x2+t/2=18, x+y = Q. (3) x2+y2_18, z-\-y=8. The diagram shows that the roots of (1) are (3,3) or (-3,-3); of (2) are (3,3) or (3,3). The roots of (2) are equal, as the line x-\-y=Q touches the circle at the point (3,3). We might say that in this case the line meets the circle at two points which happen to be coincident. The diagram shows that the line z-\-y=S does not meet the circle at all, and there are no real values of x and y which will satisfy (3). The roots in this case are imaginary. Solve these equations by the usual methods and see if the results agree with the diagram. u \ ' S 7 ~ 7 \ \ ■i- / N (\^ / SiSTcP / 1 t y N-^V\ / A (J ''! ^ / \ 'V- c(i y r ^ \ / ^ \ \ > (' .( / \ \ X ^^i / 7 s S, , / / > \ / / \ / \ P,o; y / r y 290 ALOE BRA BXSRCISB 186 /I, On the same sheet draw the graphs of the circles whose equations "are x^+y^=^'i, x^+y^=9, x^+y^=^lZ, x^+y^=3i. 2. Solve graphically x^-{-y^=l3, x—y=l. 3. Find graphically the positive integral roots of a;2+2/*=25 and 2x+3y=l8; x^+y^^lO and 2x—y^5. Approximate to the other roots. 4. The sum of two numbers is 8 and the sum of their squares is 25. Show, graphically, that this is impossible. Is it impossible if the sum of the numbers is 7 instead of 8 ? BXBROISB 186 (Review of Chapter XXII) dy Solve x + 2/ = 28, X*- 2/* = 336. 2. Solve 5x-2i/=I2, 25x»-42/* = 96. 3. The sum of two numbers is 10 and the sum of their squares is 58. Find the numbers. 4. Solve 2x-3j/ = 4, x» + 2/* = 29. 5. Solve 3x-42/ = 4, 2x* + 3xi/ = 56. 6. The sum of two numbers is 5 and the sum of their reciprocals 18 g. Find the numbers. 7. Solve x*-\-xy+2y* — 2x~7y + 5 = 0, x + y = 3. 8. Solve x* + xy—62/* = 0. x*-|-3x)/ — 2/* = 36. 9. A field whose length is to its breadth as 3 to 2 contains 664 square rods more than one whose length is to its breadth as 2 to 1. The difference of their perimeters is 60 rods. Find the dimensions of each field. 10. Solve x* + 2x2/ = 55, x2/ + 2?/* = 33. 11. Solve 2x*+3x?/ = 8, y'-2xy = 20. 12. The area of a rectangle is 300 square feet. If the length is decreased by 2 feet and the width by 3 feet, the area would be 216 square feet. Find the dimensions. 13. Solve x(x4-t/) = 150, 2/(x + 2/) = 75. 14. Solve x(x — 1/)= 15, 7/(x-(-y)= 14. 15. Sodding a lawn at 9 cents a square yard coetB $108. If it had been 10 yards longer and 6 yards wider the cost would have been half a« much again. Find the dimensions. SIMULTANEOUS QUADRATICS 291 16. Solve x3-i/»= 126, x»4-a^ + j/» = 21. 17. Solve x» + 3an/-6y*4-2a;— 2/=12, x + 2/ = 7. 18.* If (x4-t/)*-7(x4-J/)+12 = and x*t/*-6x7/ + 8 = 0, find the values of x + 2/ and xy, and thus solve these equations for x and y. 19. The product of two numbers is 28 and their difference is 5 Find the sum of their squares, without finding the numbers. 20. Solve 8x3 + y» = 280, 2x-fy=10. 21. Solve 2/ = x+ a/2, x* + 2/«=l. 22. Find two positive integers whose sum multiplied by the greater is 192 and whose difference multiplied by the less is 32. a* fe» ab 23. Solve — + — = 10, — = 3. X* y* xy 24. If 12x»-41xi/ + 3oj/* = 0, find the values of -• 25. The product of two numbers is 6 and the difference of their squares is 5. Find the numbers. 26. Solve ^ + - = 6, x-w = 4. y* y ^ \~. Solve (x + 2/)(x+22/) = 300, - + ^ = 3. y ^ 28. A regiment consisting of 1625 men is formed into two solid squares, one of which has 15 more men on a side than the other. What ia the number on a side of each ? 29. 1 2 Solve - -( — X y 1 4 = 8, -. + , = 40. X* ^ y« Solve - - - X y 1 4 6 5 12' x« + y»~ 12 30. 31. The difference of two numbers is 15 and half of their product equals the cube of the less. Find the numbers. 32. Solve x«-r8!/*=37, X2/=10. 33. Solve x + -=4, m-- = 3. y " X 34. Two men start to meet each other from towns which are 25 miles apart. One takes 18 minutes longer than the other to walk a mile and they meet in 5 hours. How feist does each walk ? u 2 292 ALGEBRA 36. Solve -+- = -, — . -4 — i = Tr' 36. Solve (x + r/)»-x-y = 20, xi/ = 6. 37. The difference of the cubes of two consecutive odd numbers is 218. Find the numbers. 38. Solve x« — x«2/* + 162/* = 28, x* + 3x2/ + 42/»= 14. 39. Solve x'+2/ = ?/* + x=3. 40. The diagonal of a rectangle is d, and the difference of the sides is s. What are the lengths of the sides ? Apply the formula thus obtained to find the sides of a rectangle whose diagonal is 13 inches, and one side is 7 inches longer than the other. 41. Solve 9x* + i/* — 21(3x + 2/)+ 128 = 0, xi/ = 4. (Make the first equation a quEwlratic in 3x + i/, by adding to 9x* + ?/* what is necessary to make a complete square.) 42. Solve x* + 4?/»-18x-36i/-)- 112 = 0, xi/ = 8 43. Solve x»+3/3= 126, x«2/ + x2/* = 30. CHAPTER XXIII INDICES BXERCISB 137 (Oral) 1. What are the values of 3 2, 2^, 1*, P", 0^ ? 2. Simplify 3 X 22 ; 3x10'; SxO"; 03-f-4. 3. When x=10, what are the values of : x3, 6x2, 200^x, 500-^x2, Gx^-fx^ ? 4. Givethevaluesof(-l)2, (-1)3, (-1)4, (-1)37^ (-1)'^ 5. What are the values of (—2)2, (—2)5, (— 2)« ? 6. Find the difference between 2^ and 32, 2^ and 52. 7. What does x* mean ? How many factors are there in x^xx* 1 8. Express in the simplest form a2xa3xa*. 9. How many factors will remain when x^ is divided by x' V What is the quotient ? 10. What are the values of : x*^x3, x^°^x* X8 O20 ^;.3 a*b^^ x' 0^° Trr a^b^ 11. What does (a2)3 mean ? Read its value without the brackets. 12. State the value in the simplest form of : (X2)2, (2/3)2, (^3)3^ (^3)4^ (^2)10. /a\3 13. What does (06)* mean ? What does ( t ) mean ? Read their values without brackets. 14. Express as powers of 10 : 100, 1000, 10,000, 10 X 100, 10^ x IO2, 10«-rlO». 298 29* ALGEBRA 15. Simplify (-l)»x(-l)3x(-l)*; (-a)V (-a)*x(-a)' to. What is the value of x if lO'^lOOO, 2^=16, 5'= 125, 3'=8I ? 17. Express 32,794 in descending powers of 10. 202. Definitions of a"'. As a^ is the product of three factors each equal to a, so a'" is the product of m factors each equal to a. a^" a.a.a . . . to m factors. Here it is understood that m is a positive integer. 203. The Index Laws. We have akeady seen that : (1) a^xa*=a^^^=a\ (2) a'=^a^'=a'^'^^a^. (3) (a2)3=a2^3=a«. (4) {ah)* = a*b*. /a\3 a' Let us now express these statements in general form, using letters to denote the indices. (1) n"'ya"=a"'*". (2) n"' — a"=a"''". (.3) {fi"')" = (i'"". (4) {ffhy"^a'"b*>^. These are called the index laws. The letters m and n represent any positive integers, and in (2) ?n>n {m is greater than n), to make the division possible. The laws, as stated in the general form, may be proved as in particular cases. INDICES 295 204. Law I. Law for Multiplication. a'"xa"=a'"^". By definition, a'" = a . a . a , . . to m factors. 0:^=0 . a . a . . . to n factors, o" Xo'' = (o . a . o . . . to m factors)(o . a . a . . . to n factors). = a . a . a ... to (m + n^ factors, = 0"*+", by definition. Also, a'" xa"xai =0"*+" x a^, 205. Law n. Law for Division, rf"^-:- re '*=«*«'»». a"* a . a . a . . . to m factors o" a . a . a . . . to n factors = a . a . a ... to (m — n) factors, if w>n, = a™~". Here the n factors in tiie denominator cancel with an equal number in the numerator, leaving m—n factors in the numerator. If, however, n>m, the n factors in the numerator cancel with an equal number in the denominator, leaving n—tn factors in the denomi- oator. when m>n, a"'-:-a" = o'"", and when n>m, a"* -:- a" = . qh m 206. Law in. Law of Powers. («'")" =a"*~. (a"')'' = a'^ . o™ . a"" . . . to n factors, = (a . o . . . to 7n factors) (a . o ... to m factors) . . . the brackets being repeated n times, =a . a . a ... to mn factors, =0™". Also, {(a"')"}p = (a'"")^=o'"'v. 207. Law IV. Power of a Product. («&)'»=«"&'*. (06)" = ab . ab . ab ... to n pairs of factors, = (a . o . a . . . to n factors) (6 . 6 . 6 . . . to n factors) Also, (a6c)"=(o6)" . c"=o"6»c". 296 ALGEBRA 208. Law V. Power of a Quotient. , , _ {l)" = l-l-l---^oniactors. to n factors 6 . 6 . 6 . . . to n factors ' 209. We have given five index laws. They are not ail independent. The second and thirH laws jnay easily be deduced from the first. (1) When m>n, a" = a'^~" xa" by Law I. .'. o"'-^o" = o'"~", which is Law II. (2) o"" xo'" = a"* "'■'" = o*'", by Law I. Similarly, a"' xa™ xa'" = a"'+"'+'" = a'"', and a"* . a"' . a"* . . . to n factors =0"'+'"+"' • • • to n ter'js=aM» (a'")" = o"'", which is Law III. For this reason the first law is frequently called the fundamental index law. EXERCISE 188 (1-18. Oral) Simplify : 1. a^xa^xa^. 2. x'xx'-fa;'. 3. (x^)*^x^. 4. (a263)2. 6. (32)2. 6. (33)2^(32)3. 7. (abf^a^b. 8. 5*^5*. 9. ((-2)2)3. 6' (-1)7 (oft)" 13. oC'xxf'Xaf. 14. a'.av.a'-v. 15. z"*-"xa;2'» + ». 16. x« + *-^x«-^ 17. (a263c*)2. 18. a:" + » . x* + ' . a5 ' ". -©'Ko^y^©*- ^- 2<' + *va;2*'*''x JC*" ■•■*"* INDICES 297 t^ \b) \c) \a) • ^' 7i>-'^ TT^-^ 7^-^ dTn + n y^ a"*P 2^ Express 4" as a power of 2 and 9' as a power of 3. 26. Divide 27^ by 9' by expressing each as a power of 3. 2nx2"-i X 22 9"x3» + 6 ^ . Simplify ^ and g^„^i • 28. Solve 52x+i_5i+3. 41=2**'; 93**s__27^"S ; 2*. 4^ . 8^=16^=^"'. 210. Fractional, Zero and Negative Indices. We have defined a'" to mean the product of w factors each equal to a. This definition requires m to be a positive whole number. Thus, the definition will tell us what a^ means, but will not tell us what a^ means, nor what a~* meaas, nor what a° means. If we wish to use in algebra such quantities as a^, a"*, a", it is necessary that we define their meanings. Now it would be very inconvenient if we gave to these new forms of indices such meanings that the index laws, alreadj' established for positive integral indices, would not apply to them. We will, therefore, give to fractional, zero and negative indices such meanings as will make the index laws valid for them as well as for positive integral indices. 211. Meaning of a Fractional Index. Since x^Y.x^=x^^'^, then if we suppose that the same law applies to fractional indices, it follows that xi X x^ = x^ "^ ^ = xi= a;. Thus, x' when multiplied by x^ gives the product x, or the square of x^ is x. But we have already represented the quantity whose square is X by Vx, 298 ALOE BRA That this is a reasonable value to attach to x^ might appear aa follows : We know that x*= Vx^, x*= Vx*, x^= Vx*, the index of the quantity under the root sign in each case being hiMpi the index in the preceding case. If now we take half of the index on each side again, it wouIq seem but natural that x- should be equal to Vx. Similarly, x^ X x^ X x^ = x* ■*" ^ "*" ^ = x. x^='^x (the cube root of x). Also, x* = Vx (the fourth root of x). and x^= y/x (the nth root of x), where n is a positive integer Thus, 4^=v/4 = 2, 125^=^125 = 5, 32^=1^32=2. By Law III, (x^)» = x*, .-. x§=-^x». (x?) =x-P, Similarly, E a 9i .". X = V xP, where p and q are positive integers We thus see that if the same laws apply to fractional indices as to positive integral indices we are led to the conclusion that x^^^vx^, when p and q are positive integers, that is, when the. index is a fraction, the denominator of the fraction indicates the root to be taken arid the numerator the power. By Law III, x« = UV =(xP)«, p ?' So that x^ means that the p"' power of the q'^ root is tc be taken, or the g"" root of the j»"' power. Thus, 8§=i?/8»=-^6i = 4, Of 8^ = (^8)» = 2» = 4. INDICES 299 It will be seen that it is simpler to take the root first and then the power. Thus, 32'^ = (\/32)» = 2» = 8. 212. Meaning of a Zero Index. By Law I, o''xa'" = o°+'" = a". Therefore, if the same law applies to zero indices as to positive integral indices, we are led to the conclusion that any quantity {zero excepted) to the index zero is equal to unity. Thus, 30= 1, (ox)''=l, (-2)»=1, (-106)0=1. 213. Meaning of a Negative Index. By Law I, Similarly, We thus see that, any quantity to a negative index is equal to unity divided by the same quantity to the corresponding positive index. Thus, a 'xa» = = a~ -1+1 = 0" = 1, a" X _ 1 O" -fXaP = -p+p = 00= 1, a' p — 1 4- = l = i-. 4» 16 27-^ ^ ^ - 1 273 (^27)« ~9 (x5)-io = a;-*= -• Since a;' = x*-;-x, x- = x^^x, x^ = x*-=rx, what would you naturally expect a;' to be equal id ? What would you expect x~^ to be equal to ^ 214. Since a"-f = — and a^ = . it follows that any af a~P factor may he removed from the numerator to the denominator of a fraction, or vice versa, by changing the sign of its index. 300 ALGEBRA Thus. -^^ = -^ = 8; -^ = — ; 4x-a3 = -^ . Ex.-SimpHfy a^/8^x V^IG^ ; {^^J^ i^S'x VW^ = 8^ X 16* = (23)§ X (2«)^ = 2»x 25=32. 16» 64 / 9a* \'g 9"' . g- 9»a«6» 27ae6« EXERCISE 189 (1-82. Oral) What is the meaning of : 1. ai 2. xi 3. 1 y-. 4. a5. 4 5. a;5. 6. x°. 7. 0-2. 8. a;-^ 9. X-*. 10. y-\ 11. w'^ 12. x-l What is the value of : 13. oi 14. lei. 15. 1253. 16. 10,000^ 17. 42. 18. 27^ 19. (i)^. 20. 5-2. 21. 10-'. 22. 1 4 2. 23. 8-i 24. (-6)0. 25. 8"i 26. (aO)-2. 27. (•25)i 28. (•16)1 29. (|)-2. 30. (-2)-«.>31. 3'',4''.f) 32. 2^.2-*. ',lj \ 35. :-;.«^36. ^.'.0i^ Write with positive indices : 33. a26-3 >I 34. 2a-3. a" Z,//:^^. 38. -V-f 39. |^:^3.V i^,-_flll ^yx-^y-*' x-^y ^ 32/r'j 3-2.t/-8 2y Writ© without a denominator : 4x. ?^. 42. ^^. 43. -^. 44. -» fc» 0-26* ' a«(c+d)-* INDICES 301 Simplify : ^* 16" i 59. Solve a;^=4, a;^=32, x*=27, x'^=Z, a;"2=8. 215. Operations with Fractional and Negative Indices. The following examples will illustrate how the index laws may be applied to the multiplication, division, etc., of quantities involving fractional and negative indices. The work will usually be simplified if all expressions are arranged in descending or ascending powers of some common letter. Thus, 5+x^+x-^2_^ -i_|_a;2 would be written in descending powers of x, thus : Ex. 1.— Multiply 2x^ + 3— x-i by Sx^ — 2— 5x-i. Ex. 2. — Divide a—b by ah—h^. (1^ (2) 3a;i_2 — ^x"^ a — a^b^ &x +9x^- 3 -4x^- 6+ ix'^ -10-15x~^ + 5a;-i 8a; +5x^-I9-13«"* + 5a;-i -ah^ -b -b 302 ALGEBRA Ex. 3.— Find the square root of 9.r— 12^*4- 10— 4.r-i4-x-i. 9a;-12x* + 10-4a;"'^+a;-i I 3x^-2 + x'^ 9x ^-2 I -12a;^+10 -12x^+ 4 6x^-4 + x'^ 6-4x'^ + x-^ 6 — 4x ^-j-x'^ Verify by squaring 3x^ — 2-\-x ^ by the method" of art. 93. Also check by putting a;=l. EXBRCISB 140 Multiply : 1.* a;2+3, a;^-2. 2. x+x^ + l,z^-X, 3. x-^-x+x^-1, x^+l. 4. 3x-2a;^+5, x~2x^ 5. a^_l-f-2a~i o2 + l-2a"i (J. {a-a^ + l)^. 7. x+5xi+6xKx^-l-x~^. 8. (x^ + 2)*. 9, x+x^^+y, x—x^y^+y. 10. (a- — 1)^. Divide and verify : 11. a+5aM+66 by a^+2bk 12. a;3— a;2+a;— 2 — 2a;- 2— 2a;-8 by a;«+24-2a;- *. 13. x^-\-x*y^-^y^ by z^+^^^J/ +2/ • 1 1 " 14. 1— 5a;"— X by 1— x^+3a;^, as far as four terms. Find the square root and verify : 15. a + 6a^+9 and 25x2— 10+a; ^ 16. a2+4a2-f6a+4o^+l. 17. 4a;^'-20a;3+37a;-30a;^+9«i INDICES SOJ» 18. 49-30x^-24a;~^+25x34-l6z"i 19. Show that °'-°'^^+^^ = (a^+fe^)-'. ^^ 'Divide x^-—z~- by x-— a; 2. 21. Divide 10a3'"_32a"'— 27a2'" + 14 by 2a'"— 7. •12. SimpUfy (a;+xi+l)2+(a;-a;^ + l)2. ^3. Simplify (Vo+l)(\/o-l)-(\/3a+\/2)(V3a-V'2). 24. Find the square root of x^—AxVx+\0x—\2\/x+9. 216. Contracted Methods. The following examples will illustrate how contracted methods may be employed. Ex. 1. — Multiply x+z^—4 by a;4-x^+4. If x + x- be considered as a single term, the product = (x + x2)«-4», [{a + b){a-b) = a*~b*.'\ = a;* + 2x^+x-16. Ex. 2.— Divide a+b by a^+bK Since a is the cube of o^ and b of b^, this is similar to dividing 2;3 + j/3 by x + y. Since {x^ + y^)~(x-ry) = x* — xy + y*, so (a + b)^{a^ + b^) = a^-nh^-irbK Ex. 3. — What is the cube root of This is evidently the cube of a binomial whose first term is 2x' and last term is — 3t/^. .'. the cube root is 2x* — Sy*, if the given expression is a perfect cube. Check when x = y=l. Usir^ the method of art. 155, the cube root of more complicated expressions may be found. 804 ALQEBRA 217. Factors with Fractional or Negative Indices. If we are permitted to use fractional or negative indices, many ex- pressions may be factored which were previously considered algebraically prime. Ex. 1. a — b 3iay be written as the difference of two squares, thun (a^)«-(6i)». .-. a-6 = (a^ + 6^)(a^-6-). Ex. 2. 3a5— 1 — 2x~^ niay be factored by cross multiplication The factors are (Sa;- + 2a;'^)(x* -x'-). Ex. 3. x*-\-xy-\-y* is an incomplete square. It may be written {x-\-y)* — {x-y^)*. .'. the factors are {x + x^y^ -\-y){x — x^y^ -\-y). EXSRCISE 141 Use contracted methods in the foUowing : yV'i.* Multiply x^-2 by x^ + 2 ; a^—l)^- by J+fet 2. Multiply a^—l+a"^ by a^ + l+o~i 3. Find the square of x— x- — 1 and of 2a— 2— o~^. 4. Write down the cube of a- + l and of 1— x-. 11 11 5. Multiply z+x-y^+y, x—x^y^+y, x^—xy-\-y\ 6. Divide x-{-y by x* —x^y^ -\-y^K 7. Divide a+2aM+6— c by a^-\-lh—c^. 8. Find three factors of x^—y^. 9. Find a common factor of a+a^b^ —2b, a—b. x+x^-6 a-b a'^+ab+b^ 10. Simpufy r — . -j j^. ::z- ^- x—5xi+Q a3-63 a+y/ab+b INDICES 306 3 jL 11. What is the cube root of a;2_6x-i-12a;^ — 8, and of x3-3J + 6a:2-7a;^ + 6x-3a;5 + l ? 12. What is the square root of 4x-*-fl2-?-3_|_ 93.-2^ and of x2+4a:+2-4a;-i+a;-2> EXERCISE 142 (Review of Chapter XXIII) 1. State the index laws. 2. Explain how meanings are assigned to such quantities as /S.* When x= 16, t/ = 9, find the values of : jk. Find the numerical values of : 8^, 9~', i/\2b*, 32-*, 16-i«. -25"^ (-64)"^. Jb. Show that i'%* X ^^8^= 108. ^6. Simplify 32"^-i-(^i.)J ^nd 812^^(iVrV- ^/7. Find, to two decimals, the value of 10* when x= i, 1, i^ |. J&. Simplify 2*+\0'>-'^^-{l)'^ + {)*^{y/J^)-^. J9. Find, to three decimals, the value of (3*)~^x v'27. 710. Simplify 16^+ 16'- 16"*- 16"^ and 32=-32*+32'» + 32~*. JlI. Simplify 5^ X 5^ X 5^ x 16^'' X 16" x 16"". yl2. Solve a;^ = 8, 2^ .4^ = 64 13. SimpUfy 4^x 6'*x -^'3 and {83 + 42)x 16"^. ^A e- 1* 2"+! 6-" , 3.2''-4.2"-i 14. Simphfy —-5 X jgr-„ and ——^-g^^. 15. Reduce to lowest terms : o+SVo-f 15 3x^+5x^+2 a^+ab a+7\/a4 12' j.*+l ' °*-*' ' 506 ALOEBKA 16. Multiply x^y^-2xy + ^x^y^ by x^ + 2y^ 17. Miiltiply x' y - -\-\-\-x~^y^ by x^y~^ —\-\-x~^y^. 4 3-8 -■• 18. Divide x* — y-* by x^-\-x^y ^ -\-y ' and o^+1-f.a"^ by a^+]+a"i 19. Divide a*"'-6*'" by a"'-6'". 20. The dividend is J/-+22/* — St/— 2, the quotient is J/-— i/'-l, the remainder is 3i/^— 1. Find the divisor. 21. Find the square root of {x + x~^)* -4(x — x~ij. 22. What is the cube root of ia»*-ia»^6i' + §a^6*-'- j'-fc^j' ? 23. If x = a»+l and 2/ = a->+l, show that ^ + ^~y = o». xy—x+y 24. Simplify -008^ 1-728^, 2-25»s, -0625'^. 25. If x + y = a^ and x~y = a ^, find the values of xy and x*+v* in terras of a. 26. If 2a = 2^-t-2-^ and 26 = 2^-2"^ find a*-b*. 27. Find the square root of (e*— e~')*-f-4 and of x*-'ix^y^+l0x*y-l4:x^y^+l3xy*-6x^y^+y'. 28. Simplify -^ x — -r- -i c b ''■ 29. Factor x* — y, x — 5x- — 6, x— 1, 4a— 6* and x»-4x+10-12x-i + 9x-». 30. If 10-30103 = 2, find the value of IQ-soJoe ^nd IQi-Bosii. 31. If 7*<»i«« = 50 and 7«<'6»« = 55, find the value of T*"'"'. 2n+l /2r.-l)«+l 32. Simplify y^^^^ x ^„.;, • INDICES 307 33 Solve 3'+ 1 + 2" = 35, 3'^ + 2''+« = 41. 34. Divide x-2{x^-x'h + 2(x^^ -x'^-^'^ by x^-x'K 36. Show that x^'^^={xVx)'' is satiafied by x=2l. 36, Find the square root of U2Vx)^-2x^+x+4x*+ Vi-i- ^. 'ii 2 CHAPTER XXIV bURDS AND SURD EQUATIONS 218. In Chapter XVIII. we have ah-eady dealt with elementary quadratic surds It was there shown by squaring that VaxVb = Vab. We might now deduce it from the index laws. From Law IV, (afe)" = a"6". Letn = i, .-. {ab)^ = ah^\ Vab=VaxVb. L L L Similarly, (a*)"=o"6", .-. \^ab=VaxVb. 219. Orders of Surds. We have already defined a quadratic surd as one in which the square root is to be taken. A cubic surd is one in which the cube root is to be taken. When higher roots are to be taken as the fourth, fifth . . . n"", they are called surds of the fourth, fifth . . . n}^ orders. 220. Changing the Order of a Surd. A surd of any order may be expressed as an equivalent surd of any order which is a multiple of the given order. n Thus, \/x=x^=x^=x^ = x^'\ \/x=-\/x*='\/x^= Vx". n Similarly, x^ = x'' = x^ = x^'', •08 SURDS AND SURD EQUATIONS 309 221. To Compare Surds of Different Orders. Any two surds may be reduced to surds of the same order and their values compared. Thus, to compare the values of V2 and "^3, v/2 = 2^ = 2« = •v/2» = v^8. if/3 = 3* = 3^ = V''3*=\5'9. It is thus seen that -^3 is greater than V2. 222. Changes in the Form of Surds. Any mixed surd can be expressed as an entire surd. Thus, 2^5= ^2» X ^5= -^40. /m — n I , .. m — n /— ^ ^ TO + n). 1 — = . / (to + n)* . — - = Vto» — n». ^TO + n 'V^ TO + n Conversely, ^250 = -^125 x ^2 = 5^2. ^I'=^fil = ^^x ^Too=i^Ioo. iJ^^^8T=^^^7x^3^ -3-^3, BXBRCISEl 148 Express as mixed surds : \ 1. ^27", A/lOOa \/562, VSo^ft, V32^, V363a«62. 2. -^16, -^SaS^Six*; -^125^, ^-"8la3,-^^=^|a*: 3. v'32, -^243; -t^J^, ^^64, V8a;2+16xy+82/2. y Express as entire surds : 4. 2V3, 10^2, 3Va, o\/5, a6V6, (o— 6)\/a— 6. 5. 2^3, 3^7, -2^16^ a^S^fe, -^^, 2v^5. 6. (a+6)y«^. (TO+«)yzLi. ^y^. 310 2 "^ Z^^ ALGEBRA 1 .* Reduce ^2, V^S to surds of the same order. Also reduce ^2 and V^S ; \/2, f 3 and V^S. 8. Which is the greater: 3\/2 or 2\/3 ; Sv^G or 7\/3 ; V"^ or v'^lO ; 1-26 or \'/2 ; ^^3 or y/51 Reduce to like surds and simplify : 9. \/8 + \/l8+V98. 10. ^500+^80-^20. 11. 3V32+5\/50-iV'r28. 12. '^l6--^r28+v^^250. 13. 1^96 -2^ ^^12 + ^'324. 14. -^32+ -1^162^+ \/l250. 15. \/75-3\/l2+5V'300+2v/48-7Vl47+3\/i. 16. x\/x^-y-ir\/x^+xhf-\/{x-\-yf-V{x^-y^){x-y). 17. Express as equivalent surds of a lower order : \/9, v/l25, \/^, \/l6^6, A?^32. 18. If '^2=1-26 approximately, find the values of : ^l6. ^54, ^2000, i^J, ^^002, f^6^. ^9. Show that 2xV2xv^2xv^4=4. 223. Rationalizing a Surd Denominator. When the numerical value of a fraction with a surd denominator is required, the value is more easily obtained when the denominator is rational (art. 165). When the denominator contains only two terms, it may be rationaUzed by multiplying by its conjugate (art. 164). BXBRCISB 144 _^^ Multiply : t- r^j Oa .^ ^' "ff 1. 2^3, 3\/5. 2. ^201, -N/3ai. 3. Vi, Vxy. 4, ^4, i/5. 5. 6Vl4, JV21. 6. -^ -2, ^-4. 7. ^^=^. ^^+6, ^iH^. '^-'""^^^ 8. 's/i+2, V«-3, \/z-2, ViH-3. SDMDS AND SURD EQUATIONS 311 9. V2+V3-V5, V2+V3+V5. ' ' ^' 10. ^-1, ^a-2, i^a+3. 11. Ve-VIT, Ve+VTT. 12. (Vl8+\/l2+\/8)2. What is the simplest quantity by which the following must be multiplied, to produce rational products ? What is the product in each case ? 13. 3V2. 14. 2V5. 15. V32. 16. V64. 17. \/512. 18. ^2. 19. ^-3. 20. \/48. 21. 3-V2. 22. Va+b. 23, 3\/2-2V3. 24. aV6-6Va. Rationalize the denominator of : 25.* IS^?. V5 26. Vl- 27. VFs. 28 *+'^'. 29. (V8+\/3)-i. 30. aa ' 2+2\/2 Va2+62-6 31 «+«»-' 3 ,9 Vz+y-Vx-y Va-\-b+Vc Vx+y+Vx—y 33. Find, to three decimal places, the value of 22-:-(3\/2-\/7)(2\/2+\/7) and of {5+Vl)~{3+\/'7). 34. n X - ?.^1^ and ?/ - ^±^, find the value of x^+y\ 2+V3 2-V3 ^ 35. Simplify (2V3-\/2)3-(V3-V'2)3. 36. Simplify ^^ + i±^ and 5±^ + ^+2V5 37. Simplify 3+V5 4-V5 I + V5 2+V5 I-2V5 15+6-\/5 4-\/5 2-f-V5 38. Show that 3 — V? is a root of the equation «'— 6**— 4a;+2=U. 312 ALGEBRA 39, The three dimensions of a room are equal. If the longest diagonal from the ceiling to the floor is 18 feet, find the length of the room to the nearest inch. 224. Surd Equations. When an equation contains a single quadratic surd, and the equation is written with the surd alone on one side, the surd may be removed by squaring both sides of the equation (art. 166). If the equation contains three terms, two or three of which are surds, the operation of squaring must be performed twice. Ex. 1.— Solve l-\-Vx=Vx-h25. ' Squaring, l-f-2\/x+a;=x+25, ,-. 2Vx=24:, Vx=\2, x=\U. Verification: l+V'x=l + \/r44=13. •v/a;+25=Vl69=ia Solve by squaring in the form Vx—Vx+2^—l and in the form l = \/x+26 — Vic, and compare the three solutions. Ex. 2.— Solve 1 — \/x=a/x+25. Squaring, \. — 2^/x-\-x = x-\-2o, .P x=144. .2Vi = 24, \.y^ -Vi=12, ,p Compare, line by line, this solution with Ex. 1. The answer is the same to both, although the equations are different. We have verified Ex. 1, and we know that a;= 144 is the correct result. Let us now verify Ex. 2. Verification: 1— -v/x=l- ^144= - 11. \/x+25=V'l69=13. It is seen that our attempt at verification shows that x= 144 is not the correct root of the equation in Ex. 2. If in verifying we could say tliat \/l44 is — 12, the equation would be satisfied. But this is not allowable, as the symbol \^~ always repreeente the positive square root (art. 63). SUBDS AND SURD EQUATIONS 318 This may be explained as follows : (1) The equation \ — Vx=Vx-\-26 is impossible of solution, as Vz-{-25 is a positive quantity, and therefore l — Vx must be positive, that is, x must be less than 1. But it is evident that no value of x which is less than 1 can satisfy the equation. (2) If we square both sides of an equation, the resulting equation is not necessarily equivalent to the given equation. A simple example will show that this is the case. Let K=— 6. Squaring, .'. x* = 36. Now the equation x* = 36 has two roots -pB and —6, and is, therefore, not equivalent to the given equation. This is similar to the case in which both sides of an equation are multiplied by a factor containing the unknown. Let x = 2. Multiply by a; -3, .-. x{x-3) = 2(x~3), .-. cc*-5x+6 = 0. The equation a;*— 5x+6 = 0, which has the roots 2 and 3, is not equivalent to the given equation. 225. Extraneous Roots. Roots which are introduced into an equation b}' sTJimring or multiplying are called extraneous roots. Thus, 6 in the first equation and 3 in the second equation are extraneous roots. Refer to Ex. 4, art. 145, where reference is made to the effect of dividing both sides of an equation by a factor containing the unknown. We have already seen the necessity of verifying the results in the solution of equations. In the case of surd equations there is an added reason for verifying, for although there may be no error in the work, the root which is found may not be a toot of the given eauatica. 314 ALGEBRA ^1-.~7r '-:3c— '=^-— 3c- BXERCISB 146 Solve and verify 1-15. Reject extraneous roota : 1. \/2x^-3=0. 2. \/3x-2=2\/x^. 3. 3a;^=a;^+4. ^4. ^5x-7=2. 5. 2x^ = Z. fi. 2^3a;-25+3=7. 7. 2(x-7)^ = (x-14)i ^. Vi+Vx+5=^. 9. Vx+i5-\-Vz=9. JO. l + Vx+2==Vx. 11. VxTi+V'^+T5=ll. 12. \/4a:2+3a;-16=2a;+2. 13. Va; + 5 = -7-== • 14. —^ = 4. Va;— 3 V4x— 2 15. Vi-3^V5+1, • 16.* Vx+4-V^4=4. Vx+Z a/5-2 17. (12+x)^-fx^=6. 18. (x+8)^-(x4-3)^=2a;i ^„ Vx+16_Vx+29 6\/x-ll_2\/x+l 19. — rz: — — " *U. -77^ — z=: • Vx+4 Vx+U 3Vx Vx+6 36 21. Vx2— a2+62=x— 0+6. 22. Vx-{-Vx—9-- , _ Vx— 9 23. \/53T5-fV'x=-,-=^- 24. \/x+3+V'xHF8=2V'x. vx— 15 25. \/x+4a--\/x=2\/6+x. 26. ^x3-6x2+llx-5=x-2. , 1 „„ \/^+6+Vx=6_„ 27. 5(70x+29)^=9(14x-15)*. 28. -^-^_-_^^_3. 29. ^!r^ ^1 + ^^-^ 30. vg^+v^ ^c. V5x+1 2 -v/o+x— Va— X li€K> <1 SURDS AND SURD EQUATIONS ^ 315 226. Surd Equations Reducing to Quadratics. Ex. 1.— Solve x-\-\/x+6=l. « Transposing, /. •v/x+5 = 7 — x. Squaring, .*. a;+5 =49— 14a;+x^ .'. a;»-15a;+44 = 0, .-. (x-4)(x-ll) = 0, x = 4 or 11. Verification : When x = 4, x-f Vx+5=4+ \/9 = 7, When x=ll, x4- \/x+5= 11 + VT6=15, .". the correct root is 4. 11 is evidently a root of x — y/x-\-o =7. Ex. 2.— Solve V8x+I-Vx+I=V3i. Transposing, .-. \''8x+ 1 = -\/3x+ Vx-\- 1. Squaring, /. 8x+l = 3x+2V'3x* + 3x+x+l, 4x = 2\/3x» + 3x, '■. 2x= V'3x» + 3x", ^ y^^^ ^ -zl f^ 4x« = 3x* + 3x, ^_^i^ ^ ^.^(^^i^ot'^ x»-3x=0, x = 0or3. Here we find on verifying that both roots satisfy the given equation. Ex. 3.— Solve 2V2x+l=S-3Vx^. Solve as in the preceding and the roots are 4 or 364, neither of which satisfies the equation. Of what equation is 4 a root.' Of what equation is 364 a root.^ Ex. 4.— Solve x^-3x-6Vx^-3x—3=-2. If the surd is removed to one side, we get x»-3x-L2=-6v'x»-3x— 3. If we now square both sides to remove the surd, we will obtain an equation of the fourth degree which we cannot easily solve. 4 i ^0 316 ALGEBRA We may obtain the solution by changing the unknown from x to \ X* — 3x— 3. similar to the method employed in art. 192. , Let Vx* — 3x—3 = y, X' 3x-3 = y*, r*-3x = y* + 3. Substituting in the original equation, y* + 3~6y=-2, :. y*-Gy + 5 = 0, y = 5 or 1. .-. x*-3x-3 = 25, or x*-3x-3=l, .-. x»-3x-28 = 0, x*-3x-4 = 0, .". x=7 or — 4. x=4 or — 1. We, therefore, have four roots: 7, —4, 4, —1. V^erify each of these and show that they all satisfy the given equation. Here both values of y were positive ; if either of them had been negative it could at once be discarded as impossible. Ex. 5.— Solve x-{-y—Vx-\-y=20, xy—2Vxy=l20. From (1), Vx+y = 5 or —4. From (2), V'xy= 12 or - 10. Here the negative values of the surds are discarded, .-. Vx+2/ = 5, Vxy=^l2. x + y = 2b, x!/=144. Solving these, x = 9 or 16, !/=16 or 9. (1) (2) EXERCISE 146 Solve and verify 1-17. Reject extraneous roots : 1. x+Vx=20. 2. x-\/x=20. 3. V'3x^+V^»^=3. 4. \/3x-5-V'x-2=3. 5. 3x4-a/5x2+11+5=0. 6. 3x + 5=-v/5x2+ll. 2 „ /- 7. -7= = 5 — 2Vx. 8. 3x-2A/7a;+4=15, SURDS AND SURD EQUATIONS 317 Vz+16 V4—X V4-X Vx-fl6 11 Vx+a+Vx+b=Va—b. 12. 13. 4(x«+a;+3)^=3(2a;2+5a;-2)^. 14. 3(a;+\/2^^)=4{z-V2^^). VSx^ + 4 - \/2a;2 + i 10. ^ a;3_u 2x2— 10a;+5=i-l. V2^+5-\/x^=2. 15. 17. 19. 21. 23. 24. 25. 227. 1 V3x2 + 4 4- V2x2 + 1 ^ 7' 2Vx+3Vy=12, 3V'x+2V^=13. x+Vxy+y=28, z—Vxy+y=12. x^+xy+y^=^\, x+Vxy-^y=\^. x^-Zx^&-^/x^-Zx+%=2. 16. 18. 20. 22. + fx-1 2 ■ V 2 xy— V a-?/=30, x+y=13. x+2/+V^+^=30, a;— 2/+Va;— «/=12. = 7. vx^ Vi"^Vi~^' x_6=36. X+2/-10. Square Root of a Binomial Surd. (V3 + V2)2=3+2+2V6 =5+2\/6. (V5-\/3)2=5+3-2\/l5=8-2\/l5. (3-^2)2=9+2-6^^2 =ll-6\/2. (Va+V6)2=a+6+2Va6. (\/a-\/6)2=a+6-2Va5. The square of Va+V6 is made up of a rational quantity a+6, which is the sum of the quantities under the root signs, and a surd quantity 2\/ab, the ab being the product of the quantities under the root signs 318 ALOE BRA The form of the square of Va\-\''h will show us how we can sometimes find, by inspection, the square root of a binomial surd. >/ a-\-h-\-2Vab=y/a+Vh, \^a-\-b—2Vab = Va — Vb. Ex. 1. — Find the square root of 7 + 2\/l2. Here we want two factors of 12, whose sum is 7. They are evidently 4 and 3. _ __ 7 + 2V'l2 = 4 + 3 + 2V'4. 3. .-. '\/7 + 2V'l2=\/4 + \/3=--2+\/3. Similarly, V'l-2Vl2 = 2 - a/3. Verify by squaring 2+ VS and 2— VS. Ex. 2, — Find the square root of 14— 6V5. To put this into the form a + b — 2VcA, first change 6^5 into 2\/46, 14-6V5=14-2\/45 = 9 + 5-2V45, .-. Vl4-eV5=V9-V'5 = 3-\/5. BXBRCISB 147 (1-7. Oral) Find the square root and verify ?- _^ r-^ 1. 5+2V6. 2. 8-2\/l2. 3. 4+2^/3. 4. 6-2\/8. 5. 104-2\/24. 6. 15+2\/56. 7. 8 + 2\/7. 8. 7-W3. 9. 9+4V5. 10. 13. 16. x+y+2VFy. 15-4\/l4. 10+ VU. a—2Va—l. 11. 14. 17. 20. x+y—ly/xy. 47 + V360. 4x+2A/4a;«-l. 12. 15. 18. 21. 2a;+2V'a;2— y*. ^___^^ 20+V300>»«>^**2'' 57 -18 a/2. 19. a-6— 2V'o— 6— 1. SURUS AND SURD EQUATIONS 319 Ex. 1. — Find the square root of 56— 24a/5. 56-24V5 = 56-2\/720. Here we require two factors of 720 whose sum is 56. When the numbers are large, as here, it may be difficult to obtain the factors by inspection. When this is the case we may represent the factors by a and b and find the values of a and 6 from the equations o6 = 720, a + 6 = 56. Solve these equations by the method of art. 194 or of art. 198 and obtain a=36 or 20, 6 = 20 or 36. The required factors of 720 then are 36 and 20. .-. 56-24-^/5=56-2^/720 = 36 + 20-2^36. 20. .-. \/56-24\/5=\/36-V'20=6-2\/5. Verify by squaring. Ex. 2. — Find the square root of f + VS. 9 /g ^ 9 + 4\/5 ^ 9 + 2\/20 4"^ 4 ~ 4 the square root is „ or -^ — (- 1. Ex. 3. — Find the square root of 2V'lO+6V'2. First take out the surd factor y/2, and we get 2VT0 + 6\/2=\/2(6 + 2V5), the square root =\/2(l + \/5). EXERCISE 148 Find the square root and verify : 1. 94-42\/5. 2. 38+12\/l0. 3. 47-12\/T5. 4. 107-24\/r5. 5. 94 + 6V245. 6. 101-28\/T3. 7. 67 + 7v^72. 8. 28-5\/l2. 9. xy^2yVxf^. 320 ALGEBRA 10.* Find the valve of l^Vl6— 6a/7 to 3 decimal places. 11. Find the value, to three decimals, of the square root of ^I -. 7+4\/3 12. By first removing a simple surd factor, find the square roots of : 7V2+4\/6, 10+6V5, 7V3-12, 59\/2+60. 13. Showthat V'l7 + 12\/2 + \/l7-12\/2=6, (1) by taking the square roots, (2) by squaring. 14. Simplify v 3+'v/i2+\/49+8a/3. ^ . /- 4-2V3 15 . By changmg 2 — V 3 into -z , find the square root of 2 — V 3, also of S + V2 and of V+^Vl IG. From the result of Ex. 1, show that 94— 42\/5 is a positive quantity less than unity. 17. If a;2(14-6\/5) = 21-8\/5, find x to three decimals. 18. The sides of a right-angled triangle are Vs and 3-f2V2. Find the hypotenuse. 228. Imaginary Surds. When we solve the equation a;2=9, we obtain a;= ± 3, and we know that this is the correct result, for the square of either +3 or —3 is 9. If we solve x^=5, we say that the value of a; is ± Vs, and we can approximate to the values of the roots as closely as we wish bj'^ finding the square root of 5 by the formal method. If we are asked to solve a;2= — 9, we might say that the solution is impossible, as there is no number M'hose square is —9. This statement is correct, but we find it convenient to say if x2^-9, then ~ = ± V^^. SUBDS AND SUED EQUATIONS 321 Such a quantity as V— 9 is called an imaginary quantity, and must be distinguished from such quantities as 5, — ^, jV7, etc., which are called real quantities. We may define an imaginary quantity as one whose square is negative, or as the square root of a negative quantity. We have already seen how imaginary quantities sometimes appear in the solution of quadratic equations (art. 190). We will assume that the fundamental laws of algebra, which we have applied in using real numbers, apply also to imaginary numbers. Thus, V^^=V9xV^l = 3V^^. ■\/ — a*=Va*x V^=aV^. These examples show that an imaginary quantity cau always be expressed as the product of a real quantity and the imaginary quantity V— 1 . The quantity V— 1 is sometimes called the imaginary unit. 229. Powers of the Imaginary Unit. Any even power of v/— 1 is real, and any odd power is imaginary. Thu3, (\/^)»= - 1, by definition, .-. (V^)3=-v'^n[, .-. (V^^)«=(-i)*= + i. .-. (\/^6_(V^n:)«x v^=+a/-i, etc. 230. Multiplication and Division of Imaginaries. Ex.1. V^^x V'^=V'2 . V^X V3 . V^, = V2 . V3x(V^^)*=-V6. Note that the product iiere is — VG, not %/6. 322 ALOE BRA Ex.2. sV^xsV^^eV^xisV^i, = 90(v'^)*=-90. Ex.3. ^ =. ^,?X yZ^ = y^l = V9^3. V-2 \/2xV-l \/2 Ex. 4. (x + 2/v'^)» = x* + 3/»(V^* + 2a;y\/^, = x» — i/* + 2a:n/\/^. Ex. 5. {a + bV^l)(a-bV^^) = a*-b*iV^\)* = a* + bK 3 Ex. 6, Rationalize the denominator of . Multiply both terms by 1 -f- V — 2 and we get 3{l + V^^)_ ^ 3(1 + V32) _ J ^32 (l-\/-2)(l + V-2) l-(-2) BXBROISB 149 (1-0, Oral) 1. Express as a multiple of V^ : V—4, V — \6, V— 81, V— a* V^^Wd, V^^9x*, V^^-b)\ 2. What is the value of (V^-l)^ (V^S (V^-l)*, V^l)' ? Find the sum of : 3. V^l, V^, V^. 4. V^^, V^OOO, \/^^49. 6. 4+\/^, 2— V — 16. 6. a + V—b\ a—bV—l. What is the product of : 7. V^, V^. 8. ^-25, V^IOO. 9. V'^^^ V^P Simplify : 10.* 3V^3+2V^^5-4:V^-[2+5V'^^48. 11. (3-|-6\/^)(3-5\/^)+(5-3\/^)(5 + 3\/^). 12. (4-3\/-i)2-|-(2+6\/^)2. 13. 2-^(l-V^). 14. (-l + v/^)^(-l-V^). 15. {a+bV^)^+(a-b\-iy\ 16. Show that i( l + \/^)2=i(-l-V^). SURDis AND SURD EQUATIONS 323 17. By finding the cube of ^( — l + V—S), show that this quantity is a cube root of unity, (art. 192, Ex. 5.) 18. Are 2± V^^, the roots of x^-ix+7=0 ? 19. If a=2+3V'^ and 6=2— 3V^, show that a+b, ab and o'+ft* are real quantities. 231. Impossible Problems. We obtained the imaginary number V — 9 in answer to the question, " What is the number whose square is — 9 ? " As we have said, this is arithmetically an impossible problem. When we obtain an imaginary result in solving a problem, we may conclude that the problem is impossible. Ex. 1. — The sum of a number and its reciprocal is 1|. Find the number. Let x=the number, .*. - = its reciprocal, 2a;»-3a;+2 = 0, 3+\/9^=T6 3±V^^ Here the roots are imaginary, and we conclude that there is no number which answers the condition of the problem. In fact, it may be shown that the sum of a positive number and its reciprocal is never less than 2. Change IJ into 2^ and solve the problem. Ex. 2.— For $30 I can buy z yards of cloth at S(10— x) per yard. Find z. The total cost in dollars = x( 1 — x) = 30. .-. a;*-iOa;+30=0, 10+ V'^=20 x=--2 What conclusion do you draw ? Would it be impossible if for $30 we substitute $25 7 $20 ? y2 324 ALGEBRA BXBROISB 150 Solve and determine if these problems are possible : 1. A line which is 10 inches long is divided into two parts so that the area of the rectangle contained by the parts is 40 square inches. Find the lengths of the parts. 2. The length of a rectangle is twice its width. If the length be increased 10 feet and the width decreased 1 foot, the area is doubled Find the dimensions. Solve also when the width is increased 1 foot. 3. A man has 20 miles to walk. If he walks at x miles per hour it will take him %—x hours. At what rate does he walk ? 4. If it is possible that x(12— x) = 36+a, and a is not negative, what must the value of a be ? BXERCISB 161 (Review of Chapter XXIV) !.♦ Multiply l + \/3-V'2by y/Q-y/2. 2. Multiply \/3+ \/2 by — + — • 3. Find the square roots of: 14+ V 180, 25-4\/21, 22+\/420, ll-\/72, 12-6\/3. 4. Find to two decimal places the values of : 1 / — 7 + V3 3 \/8+V20 VB-Vs' ^5+2v6. 2V3+V8' V5+V2 ■ Solve and verify : 5. x+\/x^=\\. 6. V4x+7+\/4x+3 = 6. 7. \/z+\/z^= ,- • 8. V6x+7-Vx+2=V2Vx+i. Vx— 4 35 9. Vx+5+ Vx-IQ=- Vx+5 10. V'l^+x - Vl6-a: = , ^^ ' Vl9+» SURDS AND SURD EQUATIONS 325 11. Multiply V'2+V3+V5 by v'5+V3-V'2 and x+y + 2Vx+y by x + y — 2Vx+y. 12. When x = 2+V3, y=2—V3, find the value or 2x+y x--2y x—y x+y 13. Expand and simplify {V2a+Vb + V2a—Vb)*. Check the result by substituting a= 13,. b= 100. 14. Solve p + x—V2px+x* = q. 15. Fina the product of 2\/3 + 3v'2+ VSO and ^2+^3- VB. 16. Find the continued product of x-l + \/2, x-\-V2, x+l + VS and z+l-VS. ,^ o- vf 3\/2 2V6 , Ve 17. Simplify -7=— 77= - 77=— r + V6+\/3 V3+1 V3+V2 18. When x = a-\-Va^—\, find the values of 1 , , 1 , , 1 X x' x' 19. Express in the simplest form : A'27-\/8 + A/l7-f 12V2-V'28-6V3 and Vll + 6V'2+\/l9-4\/r2 + V5-\/2i. „^ o- .r V 12+6\/3 , /m+n . /m— n 20. Simplify ,T and . / — ■ ^ . / — 7—' V3-(-l ^ TO— n \ TO+n 21. If x=-l-f-2V'^, find the value of x«-12x. 22. Find the square roots of T+ViS and 2a— \'4a' — 4. 23. Solve 2x*-f 6x— 6-\/x* + 3x— 3 = 45. 24. SimpUfy (Vs^- ^3+ V'2)« + (\/54-\/3- a/2)» + ( V'5- a/3+ V2)* + C\/5- \/3- ^/2)*. 26. Solve 3x* — 9x+ 11 =4Vx* — 3x + 5, giving the roots to two decimal pleices. „• ^^t Va + b-{-Va—b Va + b—Va~b 26. Simplify . ■ — . , , Vo+6— Vo— 6 Va+b-^-Va—b 326 ALGEBRA 27. Show that , — ^ + , ^ = 3. V16 + 2V'63 V16-2V63 28. Show that va+Vb cannot be expressed in the form Vx+Vy unless a* — b is a perfect square. 29. =' '■'" ' ^ \ < I \ K Simplify (-^)+(^:^==) 30. Simplify (3-2\/2)'^ + (3 + 2\/2)'. 31. Find the value of x^+x*4-x-i-i when x= v'3-trV. CHAPTER XXV THEORY OF QUADRATICS 232. Sum and Product of the Roots. Solve these equations : (1) x2-lla;4-10=0. The roots are 10, 1. (2) 2.r2- 3.r- 5=0. „ „ „ f, -L (3) 15a;H26a;+ 8=0. „ „ „ -|, -^. In (1) the sum of the roots = 11, product = 10. In (2) „ „ „ „ „ = f, „ =-f. Tn CW 2 6 8 AiX \fJ / ,, ,, ,, ,, ,, jg, ,, j^. Examine the sum and the product in each case and state how they compare with the coefficients in the given equations. Every quadratic equation may be reduced to the form ax^-{-bx-\-c—0. —b+Vb^-'iac J -6— V62-4ac Ihe roots are ^ — — and ^ . 2a 2a For brevity represent these roots by m and n, -b+Vb^-4:ac-b-Vb^-4MC -26 b m-\-n = and mn = 2a 2a a {-b-^Vb^-4ac){-b-Vb^-4:ac) 4a2 (-6)2— (\/62— 4ac)2 62-62+4ac 4ac c 4a2 ~ 4a* ~4a= atf 328 ALGEBRA Comparing these results with the coefficients a, b, c in the equation, we see that : The sum of the roots of any quadratic equation, in the standard form, is equal to the coefficient of x with its sign changed, divided by the coefficient of x^, and the product of the roots is equal to the absolute term, divided by the coefficient of x^. coefficient of x Sum of the roots Product of the roots coefficient of a;* absolute term coefficient of a;* See if these two statements apply to the roots and co- efficients of the three equations preceding. The formulae for the sum and product of the roots furnish a convenient means of verifying the roots. Thus, I find the roots of 3x*-\-x—2 = to be 5, —1, but the sum of J and — 1 is — J and the product is — |, which agree with the sum and product given by the formulae. Therefore, these are the correct roots. Are the roots of 14a;»- 19x-60 = 0, V, — ? ? 233. Reciprocal Roots. If the roots of ax^-\-bx-\-c=0 are reciprocals (like § and §), their product is unity, and therefore c , - =1 or c=a. a So that any quadratic equation, in which the coefficient of x^ is equal to the absolute term, will have reciprocal roots. Thus, the roots of 6x*— 13x + 6 = are reciprocals, since their product is f or 1. Verify this by finding the roots. 234. Roots equal in Magnitude but opposite in Sign. If the roots of ax'^-\-bx-{-c^=0 are equal in magnitude but opposite in sign (like 3 and —3), their sum will be zero, therefore _- = or 6 = 0. a So that any quadratic equation in which the second term is missing will have roots equal in magnitude but opposite in sign. Thus, 2a;* — 9 = and ox*— c = have such roots. Verify by finding the roots. THEORY OF QDADRATIC8 32^ EXERCISE 152 (Oral) State the sum and product of the roots of : 1. x2— 7a:+12=0. 2. x2_5a._ii^0. 3. a;2+6a;+l=0. 4. 2a;2_i0a;+6=0. 5. 3a;2-12a;-7=0. 6. 4a;2-17x+4=0. 7. ax^—bx—c=0. 8. ax^—{b-\-c)x-\-a=0. 9. px^—q=0. 10. ax^+a=0. 11. 3x^-4:z=6. 12. (ai-b)x^-x+a^-b^=0. 13. Which of the preceding equations have reciprocal roots V Which have roots equal in magnitude but opposite in sign ? 14. Are 4 and 5 the roots of x2-9a;+20=0 ? 15. Are S+V2, 3—V2 the roots of x^—6x+7=0 ? In which of the following are the correct roots given : 16. x2-7a;J-10=0; 5, 2. 17. x-2+3a;-28=0 ; 7, -4. 18. a;2-13a;4-36=0; 4, 9. 19. x2-12x+27=0 ; 4, 8. 20. x2-4a;-5=0 ; 5, 1. 21. 2a;2-5a;+2=0 ; 2, J. 22. In solving a;^— 2a;— 1599=0, one root is found to be 41. What must the other be ? 23. How would you show that 1-125 and 2- 168 are the correct roots of x2-3-293x+2-439=0 ? 24. If the roots of Qx^—lOx-^a=0 are reciprocals, what is the value of a ? 25. If the roots of jnx^—{m^—9)x-'rm^=0 are equal in magnitude but opposite in sign, what is the value of m ? What would then be the product of the roots ? 235. To form a Quadratic with given Roots. First Method. In the equation x^-^-px-^-q^O, the sum of the roots is —p, and the product is q. Since every quadratic equation may be reduced to the form x^-}-px-]-q=0, by dividing by the co- efficient of x^, any quadratic equation may be written thus ; X'^—X (sum of roots) + (product of roots)=0. 330 ALOEBRA If the roots are given, the equation can at once be writi^a down. Thus, the equation whose roots are 3 and 5 is x* — a;(3 + 5) + 3 . 5 = 0, orx»-8x+15 = The equation whose roots are 2-}- \/3 and 2— VS is x»-x(2+V3-|-2-V3) + (2+\/3)(2-\/3) = 0, or x*-4x+l = 0. The equation whose roots are a + b and a — b is x* — 2ax-f-a* — 6* = 0. Second Method. The equation whose roots are p and q is {x—p){x—q)=^Q. The equation whose roots are 3 and 5 is t-^ — 3)(x — 5) = 0, or x*-8x+15 = 0. The equation whose roots are § and —J is (x — §)(x+|) = 0, or (3x-2)(4x+3) = 0. The equation whose roots are 2— \/3, 2— \/3 is (x-2- V'3)(x-2+\/3) = 0, or (x-2)*-3 = or x*-4x 4-1 = 0. Either method is simple enough to apply, but the first is probably easier when the given roots are not simple numbers. The second method may be applied to form an equation with any number of given roots. Thus, the equation whose roots are 2, 3, —5 is (x-2)(x-3)(x + 5) = 0, or x3_ 19a; +30=0. EXERCISE 153 (1-16, Oral) State, without simplifying, the equations whose roots are : 1. 3, 7. 2. 3, -7. 3. -3, 7. 4. -3, -7 5. ii 6. i, -1. 7. -h-\. 8. I, J. 9. a, a. 10, —a, —b. 11. 3, 0. 12. 0, m. 13. 3,4,5. 14. 2,3,-1. 15. a,b,c. 16. a, 6, 0. Reduce to the simplest form the equations whose roots are : 17.* m+n, m—n. 18. 2a— b, 2o+6. 19. 3+V3, 3-A/a 20. 1|, -2i. 21. -2, -4, 6. 22. i, \, \. 23. Show that 1-25 and 4-64 are the correct roots of x00a;''-589*+580==0. THEORY OF QUADRATICS 331 24. Construct an equation in which the sum of the roots is 7 and the difference of their squares is 14. 25. Form the equation whose roots are a and b where a^-\-b^=2bi a-\-b=l. 26. Form the equations whose roots are a-\-b a—b ,_ -^, -— ;; i(4±\/7). a—b a-^b - 27. Find the sum and the product of the roots of : (1) (x-2)2=5x-3. (2) {x-a){x-b)=ab. (3) x{x-p)=p{x-q). (4) (2-+a)2+(a;+6)2=(a;4-c)2. 28. Solve x* — 21x2— 20x=0, being given that one root is 5. 29. If one root of a;^— 12x+a=0 is double the other, find the roots and the value of a. 30. If one root of x^-\-px-'ri&—Q is three times the other, what are the values oi p 1 236. Functions of the Roots. When m and n are the roots of ax^-\-bx-\-c=0, b c m-\-n= , mn = - • a a Here it will be seen, that the sum and the product of the roots do not contain surd expressions, while the separate roots do. If we wish to find the sum of the squares of the roots, we can do so in the following way : m^+n^ = (m-\-n)^—2mn. 62 2c 62_2ac It can also be found by taking the square of each root and adding the results. Find it that way and see if you get the same result. 332 ALGEBRA Ex. 1. — When ni and n are the roots of a,r2-|-6.T-f c=0. find the values of - + -. - -| — , m^-i-n^, m~n. m n n m 1 1 _ m-\-n _ b _ c _ b m n mn a ' a c 6»_ 2c f>^ , ''^ _ w* + n* _ (m + n)* — 2mn _ a* a b^ — 2ac n m mn mn c ac 3 , 3 / , N3 o / , ^ fe» 36c 3a6c-6» m^ + n^ = (m4-tiy — 3m,n(m + n) = — — H — „-= ^ t a^ a* a' or m,^-^n^ = (7n4-n)(m* — mn + n*) = {??!+ 'i)l(w + n) *—3mn} = etc. / .\. / V. „ 6* 4c 6*— 4ac (m — n)' = (m + n)* — 4wn = — . = s — > a* a a* Vfc* — 4ac m — n = H . ~ a The same two values of the last expression might have been found by simple subtraction, the sign depending on the order in which the roots were taken. Ex. 2. — If m and n are the roots of x^-\-px-\-q=0, find the equation whose roots are m^ and n^. Here m-\-n= —p and mn=q. The sum of the roots of the required equation is m* + n* = (m + n)*— 2mn=p*— 2g. The product of the roots = m*n* = g*. .•. the required equation is x* — x{p*—2q)-\-q* = 0. Ex. 3. — Find the equation whose roots are each greater by 2 than the roots of 6a;2— 13x— 8=0. Solve the given equation and the roots are 3, —J. .'. the roots of the required equation are V, '• .'. the required equation is x* — {i^+i)x+'-,*. 1 = 0, or 6x»-37a;+42 = 0. We might have solved the problem without finding the actual roots of th«) given equation. THEORY OF QUADRATICS 333 Let p and g be the roots of 6a;*— iSa;— 8 = 0. Then P + <1=V- and pq= — t- .'. the sum of the roots of the required equation is p+2 + g+2=p + g+4 = i5»- + * = V-. and the product = (p + 2){q+2)=pq+2(p + q) + 4,= -i + -^^+i = 7. .'. the required equation is x*-^x+7 = 0, or 6a;*-37x + 42 = 0. When would the second method be simpler than the tirst ? Ex. 4. — Find the equation whose roots are the reciprocals of the roots of mx^-^7iz-\-k=^0. Let p and q be the roots of the given equation, n , k then P + q = and pq = — • m m The roots of the required equation are - and - • p q Find the sum and product of - and - in terms of m, n and k and p q complete the solution. Compare the new equation with the given one and see if you could not write down, mentally, the equation whose roots are the reciprocals of the roots of any given equation. 237. The following method will be found useful in solving such problems as the three preceding. Ex. 1. — Find the equation whose roots are each greater by 5 than the roots of 4a;2— 5x+7 — 0. Let y be the unknown in the required equation. Then y = x + o or x — y — 5. Substitute x — y—5 in the given equation, and the required equation is 4(2/ — 5)*— 5(i/ — 5)-f 7 = 0, or 42/»-40y+100-5y + 25 + 7 = 0, Of 4j/»-45y+ 132 = 0- S34 A WE BRA Ex. 2. — Find the equation whose roots are the squares of the roots of ax^-\-bx-{-c=0. Let y be the unknown in the required equation. Then y — x* or x=+Vy, .'. the required equation is a{±Vy)* + b(±Vy) + c = 0, or ay-\-c= '+bVy, or a*y* + 2acy + c* = b*y, or a*y*^y(2ac — b*) + c* = 0. Solve Ex.'s 2, 3, 4 proceding, by this method. BXEROISE 154 1.* If m and n are the roots of a;^— 5x+3=0, find the values of 1 , 1 m , n , , - -| — , — I — , m^-j-mn-j-ri'. m n n m 2. Find the sum of the squares of the roots of x2-7x + l=0 and of Sa;^— 4x+5=0. 3. li p and q are the roots of 3x^+2a:— 6=0, find the values of - + -, V^'P'-P?+9'- q p' p« g2 4. Find the sum of the cubes of the roots of 2x2— 3a;+4=0 ^nd of z^—x+a=0. 6. Find the equation whose roots are double the roots of a;*— 9x4-20=0, (1) by solving, (2) without solving. 6. Find the equations whose roots are each less by 3 than the roots of (1) x2-llx4-28=0, (2) x2-x-l=0. 7. Find the equations whose roots are the reciprocals of the roots of (1) 2x2+x-6=0, (2) x^-pz+q=0. 8. If m and n are the roots of 3z''— 2x+5=0, find the equationfl whose roots are : (1) i and - , (2) - and - , (3) m' »ad n». n n n m THEORY OF QUADRATICS SP 9. Find the sura of the squares and the sum of the cubes of the roots of x'^-\-ax-{-h=0. 10. Find the equation whose roots are the squares of the roots of x^-\-px—q=:0. 11. Find the equation whose roots are each greater by h than the roots of ax^-\-hx-\-c=0. 12 Find the equation whose roots are the reciprocals of the roots of x^-\-x=\. 13. If m and n are the roots of x^—px-\-q=Q, show that m-\-n and mn are the roots of x^—x{p-\-q)-\-pq=0. 14. Form the equation whose roots are m and n, where m2+n2=20, w+7? =— 6. 16. If m and n are the roots of x'^-\-px-\-q=0, show that m-{-2n and 2m-\-n are the roots of x^+3px-\-2p^+q=0. 16. If p and q are the roots of ax^-i-bx-'rc=0, find the value of p*-\-p^q^-\-q* in terms of a, b and c. 238. Character of the Roots of a Quadratic Equation. Solve the equations : (1) x2— 6.r+ 9=0, the roots are 3, 3, (2) 6a:2+ a:- 15=0, „ „ „ |, -». (3) 5x^-{-lx- 2=0, (4) 2a;2-3x+ 2=0, -7±V89 10 3±V^ In (1), the roots are equal. We might say that there is only one root, but we prefer to say that there are two roots, which in this case happen to be equal. In (3), the roots are irrational, but we can approximate to their values by taking the square root of 89. In (4), the roots are also irrational, but we can not even approximate to their values. Here the roots are imaginary, while in each of the others the roots are real. 336 ALGEBRA These statements might be written thus : In (1), the roots are equal, real and rational. In (2), the roots are unequal, real and rational. In (3), the roots are irrational and real. In (4), the roots are irrational and imaginary. If we examine the roots of the general quadratic equation we will see the reason why, under particular conditions, there is this difference in the character of the roots. The roots of ax^-\-hx-\-c=0 are — 6+v'62— 4ac ^ — 6— a/62— 4ac and 2a 2a From these roots we may conclude : (1) If the particular values of a, b, c are such that b^—^ac=0, then the roots are equal, for each is evidently equal to — ^ In equation (1), a=\, fe= — 6, c = 9. .-. 6»-4ac = 36-36 = 0. (2) If b^—4:ac is a perfect square, then its square root can be found exactly and the roots are rational. In equation (2), a = 6, 6= 1, c= — 15. .-. 6*-4ac=l + 360 = 361=l9*. (3) If b^—^ac is not a perfect square, but is positive, the roots are real but irrational. Find the value of 6* — 4ac in equation (3). (4) If b^—4:ac is negative, the roots are imaginary. Find the value of 6* — 4ac in equation (4). Hence, the roots of ax^-j-bx+c^O are real and equal if b^— 4ac=0, real and unequal if b^— 4ac is positive, imaginary if b^— 4ac is negative, real and rational if b*— 4ac is a perfect square. THEORY QF QUADRATICS ^Sl 239. The Discriminant. We see then, that we can deter- mine the character of the roots of a quadratic equation without actually finding the roots. All we require to do is to find the value of b^—4ac. This important quantity is called the discriminant of the equation ax^-{-bx-\-c=0. Ex. 1. — Determine the character >i the roots of : (1) 3x^-\-5x—U^0. (2) i2x^-2ox-\- 12=^0. (3) a;2- a;+ 3=0. (4) 2a;2-16x+32=0. The value of the discriminant (6* — 4ac) in (1) is 157, .". the roots are real and irrational, in (2) is 49, .'. the roots are real and rational, in (3) is — 11, .'. the roots are imaginary, in (4) is 0, .'. the roots are real and equal. Ex. 2. — For what values of k will 4:X^—kx-]-4=0 have efqual roots ? The roots will be equal if 6*— 4ac = 0, that is, if ^-*-64 = or if i=±8. Substitute these values for k and see if the roots are equal. Ex. 3. — Show that the roots are rational of 3mx^—x(2m-\-3n)-\-2n=^0: Here 6*-4ac = (2m+3n)*— 24/«.n, = 4m*— 127/in+9n2 = (2/n— 3n)*. Since 6* — 4ac is a square, the roots are rational. Verify by finding the roots. EXERCISE 155 (1-5, Oral) 1. What is the discriminant of x^-{-4iX-\-4:=0 ? What is the character of the roots ? 2. What is the nature of the roots of x^-ir3x-{-2—0 ? 3. What is peculiar about the roots if 6^— 4ac=0 ? 4. What kind of roots have x^—5x+7=0, x^— 6x-l-9=:0, x'—x—6^Q, a;2— 4a;— 6=0 ? 5. If the discriminant is -25, what is the character of the roota^ 7. J38 ALGEBRA Determine the character of the roots of : 6.* 23-H 5x4-3=0. 7. 3r2— 72-— 5=0. 8. 4x2+7a;+15=0. 9. 9x2-12r+4=0. 10. abx'^+x{a'^+h^)-\-ah=0. 11. x'^-mx—\=^. 12. Show that x^-\-ax-\-h=0 has real roots for all negative values Jh. 13. If 9a;24-12x+it=0 has equal roots, find k. 14. If ax-—\Qx+a=0 has equal roots, find a. 16. Show that the roots of x^—x{\-\-k)-\-k=0 are rational for all values of k. 16. If x*+2x( 1+0)4-0^=0 has equal roots, find a. 17. By solving the equation x^ — 4x4-5=t, show that if a; is real, it cannot be less than 1. 18. Show that the roots of — | 1 r = are real if X x4-a x4-o a^—ab + b^ is positive. 19. Eliminate y from the equations y=mx-\-c end?/*=4ax, and find the value of c if the resulting equation in x has equal roots. 20. If 2mx24-(57n4-2)x4-(4m4-l)=0 has equal roots, find the values of m and verify. 240. Factors of a Quadratic Expression. When m and n are the roots of ax^-^bx-\-c=0, b c m 4- n — , mn = -- a a ,'. ax^-]-hx-\-c=a(x^ 4- - a; 4- - j =a{x^— {m-\-n)x-{-inn} =a{x—m){x—n). So that, if m and n are the roots of ax^-\-bx-\-c^O, the factors of the quadratic expression ax'^-\-bx-\-c are a{x—m){x—n). THEORY OF QUADRATICS 339 We can, therefore, find the factors of a trinomial like ax^-\-hx-]-c by solving the corresponding equation. Ex. 1.— Factor 6a;2+a;— 40. Solving by formula, we find the roots of 6x* + a;-40=0 are ^, — |. .-. 6a;»+x-40=6(a;-5)(a;+f) = (2a;-5)(3a;+8). Ex. 2.— Factor 12x2-47a:+40. The roots of the corresponding equation are \, |, .-. 12x«-47x + 40=12(x-|)(x-|) = (4x-5)(3x-8). 241. Character of the Factors of a Trinomial. Since ax^-{-bx-\-c=0 has equal roots when 6^— 4ac=0, it follows that ax^+bx+c has equal factors, or is a square, when b^— 4ac=0. Thus, in 3x*-30a: + 75, 6*-4ac = 900-900 = 0. .•. 3iC* — 30a;+7o ia a perfect square when the numerical factor 3 is removed. // 6'^— 4ac is a perfect square, the expression ax^-\-hx-\-c has two rational factors, for under this condition the corresponding equation has rational roots. Thus, in 20x« — x-12, 6«-4ac = 961 = 31». 2Qx* — x— 12 has rational factors. Find the factors. 242. Surd Factors of a Trinomial. \Mien we say that a trinomial can be factored, we usually mean that it can be expressed as the product of rational factors. As we have seen, this can always be done when h'^—AiOC is a perfect square. When there are no rational roots we may use the preceding method to find surd factors. Ex. — Find two surd factors of x^— 6a;-[-4. nx«-6a;+4 = 0, a; = ^^^^ = 3± Vs. .-. a;»-6x4-4 = (x-3-V5)(a;-3+'\/6). Verify by multiplication. e2 340 ALGEBRA EXERCISE 156 Factor, by trial if you can, otherwise by solving the corresponding equations and verify : 1, 3x2-17x+10. 2. 20a;2+3x-108. 3. a;2_2.f-1783. 4. lS00a--5a-l. 5. 299^2+ lOx-1. 6. 221a:2-458ax+221a2. 7. Show that 12x^— 15x+4 has no rational factors, 8.* x-+4x— 3 has no rational factors. Find two surd factors of it. 9. If X-— 8x4-Z; is a perfect square, find k. 10. If ax^—kx-'rQa is a perfect square, when the factor a is removed, find k. 11. Express x^— 6x— 11 as the product of two surd factors. 12. Factor 144a;*— 337a;2?/^-|- 1442/*. When this expression is equal to zero, find four values of the ratio of x to y. 13. If x—2 is a factor of 120x3— 167x-—ax+56, find the value of a and find the other two factors. 14. By finding the square root of ax^+bx+c, find the relation which must connect a, b and c when this expression is a perfect square. 243. A Quadratic Equation cannot have more than two Roots. We have seen that the equation ax^-\-bx-{-c—0 has two roots, and since this equation represents every quadratic, it follows that every quadratic equation has two roots. It cannot have more than two roots. Let m and n be the roots of ax^-{-bx-\-c=^0. Then ax^-{-bx-\-c=a{x—m){x—n) (art. 240). a{x—m){x—7i)—0. Since this product is zero, one factor must be zero. But a is not zero, for the equation woukl not then be a quadratic. Therefore, either x—m=0 or X' n—O. THEORY OF QUADRATICS 341 But no values of .r other than m and n will make either of these quantities equal to zero. m and n are the only roots. Since the quadratic equation ax^-{-bx-\-c=^0 has only two roots, then the quadratic expression ax^+bx+c can be resolved into linear factors in only one way. EXERCISE 157 (Review of Chapter XXV) 1. What is the sum and the product of the roots of ax^ + bx-\-c = ? 2. Under what condition are the roots of ax*-i-fea; + c = reciprocals ? When are they equal in magnitude but opposite in sign ? 3. When are the roots of ax*-\-bx + c = 0, (1) equal, (2) real, (3) imaginary, (4) rational ? 4.* If p + 9 = 4 and pq=5, find the values of : 6. Find the sum and the product of the roots of (3a;-2)(a;-3) = (x-l)(a;-5). 6. Find the equation whose roots are each one-half of the roots of 4a;»-20x+21 = 0. 7. Find the sum of the squares of the roots of 3a;*— lla;+ 1 = 0. 8. Find the equation whose roots are twice as great as the roots of 24x*-38a;+15 = 0. 9. For what value of k will x*— I0x=k have equal roots ? 10. Find the equation whose roots are m and n when m^ + n^^l^ and mn=35. 11. Factor 5256a;*+a;-l and 221a;*- 8a;— 165. 12. Form the equation whose roots are m and n where m' + ^'=28 and m+w = 4. 13. Find the sum of the roots of {x—a)* + {x—b)* = {x—c)*. 14. Construct the equation whose roots are the reciprocals of the roots of 17x* + 53x— 97 = 0. 16, Express x* + 6a;-f-7 as the product of two linear factors. 342 ALGEBRA 16. Construct the equation whose roots are each greater by 7 than the roots of 2x*+ llx-21 = 0. 17. Find the equation whose roots are each three times the roots of ax* + 6x + c = 0. 18. If m and n are the roots of ax*-^bx + c = 0, find the equation 19. Show that {a + b + c)x* — 2x{a + b) + a + b — c = has rational roots. What are they ? 20. If one root of x*—px+q = is double of the other, show that 2p* = 9q. 21. If m and n are the roots of x*+/)x + g = 0, show that p and q are the roots of x* + x(m + /i — mn) — mn(m + n) = 0. 22. Show that the equation (mx-\ j =4ox has equal roots for all values of m. 23. Find the values of k for which the equation x* + x{2 + k) + k + 37 = has equal roots. 24. Since x* — 8x-20 = (x- 10)(x+2), for what values of x is the expression x* — 8x— 20 equal to zero ? For what values is it negative ? For what values is it positive ? 25. Show that it is impossible to divide a line 6 inches in length mto two parts such that the area of the rectangle contained by them may be 10 square inches. 26. For what values of k is 4x* — x(i+8) + A; + 5 a perfect square? Verify your result. 27. Find the sum of the cubes of the roots of x* + mx + n = 0. 28. Find the sum of the squares of the roots of x»-x(l4-o) + J(l+a + a*) = 0. 29. Find the sum and the product of the roots of a ^ _ '^ X — a X - b X — c 30. If the sum of the roots of a*' — 6x+ 12a = equals their product, find a and verify. 31. It is evident that a is one root of (x — c) (x— 6) = (a— c) (o— 6). Find the other root. THEORY OF QUADRATICS 343 32. If a;* — 5x— 3a and x*— lla;+3a have a common factor, it must be a factor of their difference. Make use of this; to find the value of a for which x'' — 5.t— 3a = and x*— lla;+3a = will have a common root. Verify by finding the roots. 33. The absolute term in an equation of the form x*-\-px + q = is misprinted 18 instead of 8. A student in consequence finds the roots to be 3 and 6. What were the roots meant to be ? 34. If m and n are the roots of ax^-\-bx-\-c — 0, show that m + n and 1 — are the roots of aca;* + 6x(a4-c) + 6* = 0. m n 35. Two boys attempt to solve a quadratic equation. After reducing it to the form x^-\-px-\-q = Q, one of them has a mistake only in the absolute term and finds the roots to be 1 and 7. The other has a mistake only in the coefficient of x, and finds the roots to be — 1 and — 12. What were the corrnct roots ? 36. Express x*-\-2hx-\-c ae the product of two linear factors in x. CHAPTER XXVI supplementary theorems and exercises Additional Examples in Factoring 244. Product of two Trinomials. If we multiply a-2b-3c by 2a— &4-c the product may be written 2a^-5ab+2b^-5ac+bc-ZcK The first three terms of the product, which do not contain the letter c, are evidently the product of a— 2b and 2a— b. The last term, — Sc^, is the product of —3c and c. If we wish to factor the product of two trinomials, we may do so by the method of cross multiplication, which we used to factor a trinomial. Ex. 1.— Factor 2a^ - 5ab-{-2b^— Sac -\-bc-3c^. First factor 2a*—5ab-{-2b*, and then choose such factors of —3c* as will give the remaining terms in the product when the complete miiltiplication is performed : 2a*-5o64-26«-5ac-{-6c- 3c» a —26 —3c 2a — b + r,. If the terms of the factors are written under the terms from which they are obtained, it is not difficult to obtain by trial the factora of an expression of this type. Ex. 2.— Factor 4a2+362_l2c2-8a6-8ac. Arrange the expression thus : 4o«-8a6 + 36*-8ac-12c* 2a - 6 -I- 2c 2c -36 - 6c Show by multiplication that these factors are correct- SUPPLEMENTARY THEOREMS 346 BXBRCISE! 158 Write, mentally, the products of r 1. a-26-i-c 2. Zx—y-\-\ 8. 3^—46+ c a~ b-c 3x+y—2 2a- b-2c 4. 3x-|- y— 4 5. o— 6+4 6. 2x— 5y+ z 2x—3y+3 2a— b 3x~-2y—Sz 7. 2a~3b-5c 8. 3m— 2n+l 9. a^—2a+3 2af36 2m— 3»+4 a^-\-3a—2 Factor and verify : 10. a^~^ab-r4:b^-a+2b-12, 11. 2x'^-i-xy—6y^+2xz+llyz~4z*. 12. 2a2-+-662~3c2+76c— 5ca— 7ao. 13. 2a;2-7a;t/-222/2-5a;+35i/-3. 14. 6a^+at>-l2b^-2a+3lb-20. 15. a;2— a;z— 6z2— 2xy+&y3. 16.* Divide the product of 6a^—5ab-i-b^i-na—4:b+3 and !i-i-6-2 by 3a2-f 2a6— 62_5a-f 36-2. 17. Reduce to lowest terms 4p'^+2lq^- 18r^+33qr+Qrp~3lpq ^p^—7pq-r3q^~2pr+3qr-Qr^ 18. If 3x-\-2y—5z is a factor of 3x2-faa;?/— 6i/2-t-6a;z4-cy3— lOz*, tt^hat are the values of a, b and c ? 19. Write the expression x^4-xy—2y^—x-{-l0y~l2 in the form z'^-{-x{y—l)—{2y^—10y-\-12). Solve the corresponding equation for X and thus find the factors of the given expression. 20. Solve x2—5ax+6a*+7a;-17a+ 12=0. (W bv factoring. {2\ by fche general formula. 346 ALOmBRA 21. Express, in the factor form, the L.C.M. of 6a2— 5a64-6--(-ac— c^ and 6«2-ffl6_262— oc+46c— 2c2. 245. Sum and Difference of Cubes. We have seen that aS + 63 = (a + 6)(a«-a6 + 6*), a^-b'' = (a-h){a'^-\-ab-\-b*). Similarly, (a + 6)» + c» = (a + 6 + c){(a + 6)*-c(a + 6) + c*; , and (a-6)3-cS = (o-fe-c)|(a-fe)« + c(a-6) + c*J. Ex. 1.— Factor a^+h^^c^—Zabc. Add to a' + 6' sufficient to make the sum the cube of a + 6, that is, add 3a»6 + 3a6^ Then a^ + b^ + c^ — Zahc, = a3 + 63 + 3a»6_^3oj»4.cS_3aJft_3a5i_3a6c, = (a + fc)» + c»-3a6(a + 6 + c), = (o + 6 + c)!(a + 6)*-c(a + 6) + c*-3afe}, = (o + 6 + c)(a« + 6* + c*-a6-6c-ca). The factors of this expression are important, and the pupi' should endeavour to retain them in memor3^ The expression is the sum of the cubes of three quantities diminished by three times their product. One factor is the sum of the three quantities, and the other is the sum of their squares diminished by the sum of their products taken two at a time. We should recognize expressions which are of the same form as this type expression. Thus, a' + 6' — c' + 3a6c may be written in the form a'' + 6' + (-c)S-3a6(-c), and it is now seen to be the sum of the cubes of a, b and — c, diminished by three times their product. The factors of a^ + ^^ — c' + 3afec may at once be written down from the factors of the type form by merely substituting — c for c. .-. a* + b^-c^ + 3abc = {a + b-c){a* + b' + c*-cih + bc + ca). o»_fe3-c8-3a6c = o« + (-6)' + (-c)»-3a(-6)(-c), = (a-b-c){a* + b* + c* + ah-bc + ca). 8x' + 27!/»-2» + 18a^2 = (2a;)» + (3j/)» + (-2)3-3(2a;)(32/)(-z), = 'Zx+3y—z)('ix* + 9y* + z* — 6xy + dyz4-2zx. SUPPLEMENTARY THEOREMS 34T Ex. 2.~Factor a^-\-h^^l — '6ah. = (a + 6+l)(a*+6»+i--a&-a— 6V Ex. 3. — Find one factor of (a:+t/)H(y+2)3+(z+a:)3-3(a;+2/)(y+2)(3+a:). This is of the same form as a* + 6' + c^ — 3o6c, where a = x-\-y, b=y-\-z, c—z+x. One factor is x-\-y-\-y-^z-\-z-\-x or 2{x-\-y-\-z). The other factor is lengthy, but is easily written down. Ex. 4.— If a+6+c=0, show that a^+h^+c^=Zahc. This is equivalent to showing that a' + 6^+c' — 3a5c = 0. Now this quantity will be equal to zero, if one of its factors is zero. But a-\-b+c is already seen to be a factor, and since it is given equal to zero, .■, a3 + 6» + c»-3a6c=0, or a3+6»+c' = 3a6c. We have thus shown that if the sum of three quantities is zero, the sum of their cubes is equal to three times their 'product. Prove this also by substituting —h — c for a. Ex. 5.— Show that (a-6)3-^(5-c)3+(c-a)3=3(a-6)(6-c)(c-a). Here the sum of a — h, h — c, c — a is zero, and therefore the result follows at once from the preceding theorem. Similarly, (a + 26- 3c)3 + (6 + 2c- 3a)' + (c + 2a- 36)» = 3(a -f- 26 - 3c)(6 + 2c - 3a)(c + 2a- 36 ), since the suro of a + 26 — 3c. 6 + 2c-- 3a, c + 2a — 36 is zero. EXERCISE 159 Factor : 1. (a+26)9-c3. 2. a'^-{h-c)\ 3. (a+6)'+8c'. 4. (aJ-6)3-t-(c+i)'. 6. {x-y)^~{a-h)\ 6. [2x-yY-\-{x-2y)K 7, (3a-6)'-(a-36)». 8. 8(3a-6)3-27(2a-3i)s. 9, a*—b^-\-c*+Zabc tO. %i^-iry^-\^z^~%xyz. 348 ALGEBRA 11. a3+6»-l+3a6. 12. l+c»-rf3+3c<f. 13.* 8x3-1/3-1 253=' -SOxyz. 14. (a4-6)HcH'-3c(a4-6l What is the product of : 15. a—b—c and a^-\-b^+c^-{-ab+ac—bc. 16. 2x-y+3z and 4xH2/^+922+2Ti/-6xz+32/a. 17. 1-a-b and l+a^+fts+a+fi-aft. 18. 2a-36-4 and 4a2+6a6+962-126+8a+16. What is the quotient of : 19. 1—a^+b^+Zab by l—a+b. 20. 27m3— n^- 1— 9mn by 3w— n— 1. 21. a3+ 12563- l + l5o6 by a24.2562+l-5a6+a+56. What is one factor of : 22. (4a+36)3-(a+26)3. 23. (a:2-3x+7)3+8. 24. (o2-3a+2)3-(a2-5a+7)3. 25. {a+b)^+{c+d)^-l+3{a^b){c+d). 26. Prove that the difiference of the cubes of 4a2+a+l and 2a2_2a-f3 is divisible by the product of 2a— 1 and a+2. 27. Show that a^+b^+c^ — 3abc is equal to ^{a+b+c)>{a-b)^Mb-c)^-\-{c-a)^\. 28. Write down a quantity of the type a^ -\-b^-{-c^—3abc, of which ^x—2y+z is a factor. What is the other factor ? 29. If aH-6— c=0, show that a^+b^i-3abc=c\ 30. If x=y+z, show that x^=y^'^z^+3xyz 31. If a+b+c=0, show that (o+26)3+(6 + 2c)3 + (c+2a)3=3(a + 26)(6+2c)(c+2a). 32. Show that (x-y)3-l-(y-2)3+(z-x)3-3(x-J/)(i/-2)(z-x)=0. 33. Show that (a+36-4c)3+(6 + 3c -4a)3+{c+3o-46)3 =3(a+36-4c)(6 + 3c-4a)(c+3a-46). STTPPLJfl,f^fJ7TARY THEOREMS 349 34. If z^a—h, y=a+b, z=2a, show that x'^-\-y^^Zxyz=^z^. 35. Find the value of a^-h^->rC^-\-'ioiic. when a=-32, 6=--46, c=-14. 36. Reduce to lowest terms : 2a2-5a6+3624-a«-3c2 ^° (a:-2?/)2+(22/-z)2+(3-a;)3* 37. Find two factors of the first degree of {ax-\-hy-\-azY-\-{hx+ay+bzY. 38. When x=h-\-c, y=c+a, z=a-\-b, prove that x^-\-y^+z'^—Zxyz=2{a^-\-h^+c^—Zahc). 39. Prove that a^+h'^+c^—ah—ac—hc is unaltered if a, 6, c be each increased, or each decreased by the same quantity. 40. Solve (a;-a)3 + (6-.T)3+(a— &)^=0. 246. Grouping Terms. We have already seen (art. 91) that we can frequently obtain a factor of an expression by 8 suitable arrangement of the terms. The following examples will give further illustrations oi this method. Ex. 1.— Factor a\h-c)-\-h'^{c~a)+c\a-h). Arrange in descending powers of a, and the expression ' = a2(6-c)-a(62-c2)~L6c(6-c), = (b — c){a^ — ab — axi-{-bc), =^{b—c)a{a—h) — c{a—b)\, = (6— c)(a — 6)(o— c). When the factors are written in cyclic order (art. 140), o*(6-c) + fe^(c-a) + c*(a-6)=-(a-6)(6-c)(c-a). This expression may also be factored by writing it in the equivalent form {a--h^){b-c)-{a~b){b-^~c^). In this form a — b and b — c are seen to be factors. Complete the factoring by this method. The expression cr(6* — c2) + 6(c^ — a^) + c(o* — 6^) differs on.y in sign from a2(6-c)+fc»(C"a) + c2(a-6), .-. a(62— c2) + 6(c2-a2) + c(a«-62) = (a-6)(6-c)(c-o). Also, o6(o — 6) + 6c(6 — c) + ca(c— o)= — (a — 6)(6 — c)(c— a). 350 ALGEBRA Ex. 2.— Factor a\h — c)+h^{c-a)^c^{a—h). The expression =a^(6 — c) — 0(6' — c') + 6c(6* — c*), = (6 — c)(a3 — afe*-aAc — ac* + 6*c + 6c*). Now arrange the second factor in powers of b, and proceed as before and obtain — (a — 6)(6 — c)(c — a)(a-r^ + c). Factor also by using the second method of Ex. 1, writing the expression in the form (a' — 6^)(6 — c) — (a — fe)(6' — c*). What are the factors of 0(6^ — c') + 6(c* — a')+c(a* — 6^), and of a6(a'-6«) + 6c(6*-c*) + ca(c«-a«) ? Ex. 3.— Factor a2(6+c)4-62(c+a)+c2(a+6)+2a5c. Arrange in descending powers of a, and the expression = a*(6+c) + a(6* + 26c + c*) + 6c{6 + c), = {6+c)(o»+o6+oc+6c), = (6+c)(a + 6)(a+c) = (o + 6)(6 + c)(c+a). Ex. 4.— Factor {a'^—h^)x^-{-{a'^-{-h^)x+ah. Expressions of this kind, when written in descending powers ot Z. are easily factored by cross multiplication in the usual way. (a»-6«)x* + (a*-6*)a; + a6 (o +6 )x +a (a —b)x +6 The factors are (a + 6)x4-a and (a — 6)x+6. BXERCISB 160 Factor and verify 1-8 : 1. acx'^+x{ad-{-bc)-\-hd. 2. mpx*-\-xy(qm—pn)—nqy*. 3. a;2(o«-62)+4a6a;-(a2_62)_ 4. (p*-g')y*+22/(p2+g2)+l)2-g2. 5. z*(a+6)+a;(a+26+c)+6+c. 6. a;*(a2_a)^a.(2a2_3a^2)+o2-2a. 7. o«{6+c)+a(6«+36c+c«)+6c{6+c). 8. o6(a+6)+6c(fe+c)+co(c+o)+3a4*' SUPPLEMENTARY THEOREMS 351 9.* xHy-z)+y^z-x)-\-z^x-y). 10. xy{x—y)+yziy—z)+zx{z—x). 11. x{y^-z^)+y{z^-x^-)+z{z^-y^). 12. a(b^-c^) + b{c^-a^)+c{a^-b^). 13. a^b*-c*)+b\c*-a*)+c^{a*-b*). 14. Divide a^b-c)^b^{c-a)+c^a-b) by a2(6-c)+62(c-a)+c2(n-6). Solve and verify : 15. abx^—x{ad-\-bc)-\-cd=0. 16. {a^-b^)x^-Aabx=a'^-b\ 17. x2(c-6)+a2(6-a:J+62(x-a)=0. 18. a6x2-x(a2+62)+a2-62=0. 19. {a^-ab)x^+{a^+b^)x=^ab+b\ 20. Find a common factor of abx^-\-x{a^—2ab—b^)—a--r-b^ and a^x^—a^x—ab—b\ 247. The Factor Theorem. Ws have already seen that any expression is divisible by x~a, if the expression vanishes when we substitute a for x (art. 101). Any expression whose value depends on the value of a; is called a function of x (art. 114). Any function of x may be conveniently represented by the symbol /(a:), which is read " function x." The factor theorem might be stated thus : f{x) is divisible by oc—a if f{a)=0. Thus, if f{x) = x^-lx^+nx-2, /(2) = 8-28+22-2 = .'. «'— 7x*+ 11a; — 2 is divisible by x— 2, If /(.c) = x3 — 4x«a+5xa*+10a', then /(-o)=-a3-4as-5a3+10a3 = 0, ,*, T*— 4x*a -|- 5xa* + 10a* is divisible by x-\-a. 362 ALGEBRA 248. Factors ol x'^ ± a**. We nave already seen that x* — a'^ = {x—a)[x-{-a^, a;*-o*={x*-a»)(x* + a2)=(x— a)(x+o)(x»+a«>- Here we see that x— a is a factor of each. Is x — a& factor of z^ — a^ ? »Vhen we substitute a for x, a;—o is a factor of a;^—o^ (1) Is x—a a factor of x"— o" ? When we substitute a for x, a;" — a" — a" — a" ST! U- «"— a" is divisible by a;— a, (2) Is x-\-a a factor of a;" — o" ? When we substitute ~a for x, a;"— a" = (— a)"— o". Now (—a)"— a" will be equal to zero only when (— o1" = cf", and tins J9 true only when n is even, X"—a^ is divisible by x+ct when n is even- Thus. a;*--a*, re*— a*, a;*— a*, etc., are divisible by x+a, but a:*— o*. *" — o*, etc., are not divisible by x-\-a. (3) Is x-\-a a factor of x"-{-a" ? Examine this, as in the preceding, and show it is a factor only wiiOD * is odd. (4) Is a;— a a factor of a;"+o" ? We thus conclude that, when w is a positive integSK, (1) x°— a° is always divisible by x—a (2) x"— a" is divisible 6y x+a when n is even. (3) x°+a" is divisible by x+a when n is octd. (4) x°+a" is never divisible by s— a. 249. Quotient on dividing a?"±«" by act©. ^2— o» a:*— a* , , , . » , , ■ 1) — = x-f-a. =sx'-\-x'aA-!ea*+a*. x—a x—a ~ - =a*4-ir'C--i-a* — ^^^^ — =af* J-a?"o+a;*o*+a»'-i-<** SUPPLEMENTARY THEOREMS 353 Verify these results by division or miiltiplication. Notice that the signs are all positive, and that the powers of x are descending and those of a are ascending. Similarly, = x^ + x^a-\-x*a*-\-x^a^-{-x*a*-\-xa^+a*, and — a;"~^+a;"~*a+a;"~*o*-f . . . +xo""*+a"~*. X — a (2) — ; — =x—a. — ■ — =x^—x^4-xa*—a\ x-j-a x-\-a Verify and note the peciiliarity in the signs. Write down the value of — - — and of ; • x-\-a x-\-a (3) — ^ — = x* — xa-\-a*. — — — =«* — x'a+a;*o* — xa'+a*. x+o x-\-a x' + a' x*"+^ + o'"+'- Write down the value of — and of ; • x-\-a x-\-a EXERCISE 161 1. If /(a;)=a;3-8a;2+19x-12, find the values of /(I), /(2), /(3), /(4), /(5). What are the factors of x^— Sx^+lGx— 12 ? 2. If /(x)=x*— 2x3— x2+2x, find the values of /(2), /(I). /(O). /(-I), /(-2). What are the factors of /(x) in this case ? 3. Prove that x^*— ?/^* is divisible by x—y and x+y. 4. Prove that x^ 2—1 is divisible by x—1, x+l> a^^+l» ^j'+l* 5. Prove that x^'-j-y^' is divisible by x-\-y and that x^+32 is fivisible by x+2. Write down the quotients in the following divisions : a;3-i-«3 a*— 6* a*—b* x'—\ 6. ^^ . 7, ^. 8. — . 9. =. x-j-y a—b a-\-b x— 1 10 * *-i+32 j^ a:*-81 ^^ ^'°-"' 13 («+fe)^-l a;+2 ' x-\-Z ' ' x^—a' ' (o+ft) + r A A 364 ALGEBRA 14. State one factor of ; x^-b^, a-'+b\ x3-64, m''+— ,, {x+y)^-l. What is the product of : 15. o^4-a*+o+l and a— 1. 16. m* — in^-^m^—m-\-l and m-\-l. 17. a^+a*b^+a^b*+b'^ and a^—b\ 18. Prove that x^+3x*+ix^+224L is divisible by x'+l. 19. Show that x+y, x^+y^, x^+y^, x* + y*, x^+y^ and a;i»+y" are factors of x^*—y^*. 20. If X— a is a factor of a;^+pa;+?» find the relation between o, p and g 21. If f{x)=Tnx^+nx-]-r, find /(a) and show that f(x)—f{a) is divisible by x—a. 22. If a;— 1 is a factor of x^—k^x^ + lOkx—10, find the values of k and verify. 23. Write down the quotient when (1) x—a is divided by x^—a^. (2) x+a is divided by x*+a*. 1 1 (3) x—a is divided by x^—a^. (4) x+a is divided by x^-\-a". 250. Symmetrical Expressions. An expression is said to be symmetrical with respect to any two letters if it is unaltered when those two letters are interchanged. Thus, x + y and x* + 2/* are symmetrical with respect to x and y, but x^ + xy is not symmetrical. Similarly, a + b + c and ab + bc + ca are symmetrical with respect to a and b, b and c, c and a, for if any two be interchanged the expressions remain unaltered. 251. Cyclic Symmetry. An expression is said to be symmetrical with respect to the letters a, b and c, if it U unaltered when a is changed to 6, 6 to c and c to a, that is, when the letters are taken in cyclic order- SUPPLEMENTARY THEOREMS 366 Thus, a* \-b^-\-c* — ab~bc — ca is symmetrical with respect to a, 6 and c, for wh"'n the letters are changed in cyclic order the result is 6* + c*+o* — 6c — ca — a6, which is equal to the given expression. The expression a^ + 6* + c' — 3o6cd is symmetrical with respect to a, b and c, but not with respect to a, b, c and d. The only expression of the first degree which is symmetrical with respect to a, b and c is a~\-b-\-c or some multiple of it as k{a-\-b-\-c). There are two expressions of the second degree, a^-\-b^-\-c^ and oi + 6c +ca,. and the sum of any multiples of these, such as k{a^-\-b^-i-c^)+l{ab-{-bc+ca), which are symmetrical with respect to a, 6, c. 252. Symmetry applied to Factoring. The factor theorem may be applied to the factoring of many symmetrical expressions. Ex. 1.— Factor a{b^-c^)+b{c^-a^)-^cia^-b^). If we put a = b, the expression equals zero, a — 6 is a factor. Since the expression is symmetrical and a — b is shown to be a factor, it follows that b — c and c — a must be factors. We have thus found three factors each of the first degree. But the given expression is of the third degree, and, therefore, there cannot be another literal factor. There may be a numerical factor. Suppose A; is a nimaerical factor, .-. a(6*-c*) + 6(c*-a«) + c(a2-fe*) = t(a-6)(6-c)(c-o). Since this relation is true for all values of a, b, c, let a=l, 6 = 2, c = 0, then l(4-0) + 2(0-l) + = Jfc(l-2)(2-0)(0-l), .-. 2 = 2k, or k=\, :. a(6*-c*) + 6(c«-a«) + c(a*-65i) = (o-6)(6-c)(c-a). In finding the vaUie of k, any values of a, 6, c may be used provided they do not make both sides of the identity vanish on substitution. A A 2 366 ALGEBRA Ex. 2.— Factor (a+6+c)3+(a-6-c)3+(ft-c-o)34-(c-a-6)3. If we put a = 0, the expression vanishes, .". o must be a factor, and, therefore, b and c. Complete the solution aa before, and show that the expression equals 24a6c. Ex. 3.— Factor a^{h-c)+h\c-a)-^c^{a-h). As in Ex. 1, show that a — b, b — c, c — a are factors. Since the expression is of the fourth degree it must have another factor of the first degree. The remaining factor must be of the form A-(a + 6 + c). .-. a^b-c) + b^(c-a) + c^a-b)=^k(a-b)(b-c){c-a){a + b + c). Substitute numerical values for a, b and c and show that the factors are -(a-6)(fe-c)(c -a){a + b + c). Ex. 4. — Simplify (a-6-2c)2+(6-c-2a)2+(c-a 26)2+(a+6+c)2. This expression is symmetrical with respect to a, b and c and is of the second degree. In the simplified result there can be only two kinds of terms, squares like a* and products like ab. The coefficient of a* in the result is 1+4+1 + 1 or 7, /. one part of the result is 7(o* + fe*+c*). The coefficient of ab is -2-4+4 + 2 = 0, .*. the complete result is 7(o* + 6*+c*). Check by letting o = 6 = c=l. Ex. 5. — Simplify (a+6)(a+6-2c)+(6+c)(&+c-2a) + (c+a)(c+a-26). The coefficient of a* in the result is 1 + 1 or 2, /. one part of the result is 2(a* + fe* + c*). The coefficient of ab is 2 — 2-2 or —2, .*, the other part of the result is — 2(a6 + 6c+ca), ;. the complete result is 2(o*-i- 6* + c* — o6 — 6c— ca). SUPPLEMENTARY THEOREMS S5l FIXBRCISB 162 (1-12, Oral) With respect to what letters are these symmetrical •. 1. a+b. 2. a+c—b. 3. x'^+y'-'rxy. 4. ab+bc+ca. 5. a^+b^+c^^3abc. 6. x^+y^+x-y. 7. 3{p^+q^+r^)-2{pq+qr+rp). 8. What is the simplest expression of the first degree which is symmetrical with respect to x and y 1 a, b and c ? a, b, c and d 1 9. What expression similar to a^-\-b^-\-Zab is symmetrical with respect to o, 6 and c ? 10. Simplify {a^by-+{b+cy+(c+a)^ and {a-b)^+ib-c)^+{c-a)\ 11. li a-{-b is a, factor of any expression, symmetrical with respect to a, 6 and c, what other factors must it have ? 12. When {a-\-b)^-\-{b-{-c)^+{c+a)^ is simplified, the coefficient of a^ is 2, of a^b is 3 and of abc is 0. What must the simplified form be ? Simplify : 13,* {a-b+c)^+{b-c+a)^+{c-a+b)\ 14. {a+b){a+b—c) + {b+c){b+c—a) + {c+a){c+a—b). 15. {x—y){px-\-py—z)^{y—z){py+pz—x) + {z—x){pz+px—y). 16. (a-6)H(6-c)^+(c-a)3. Factor : 17. x^{y-z)+y^{z-x)+z^{x-y) 18. xy{x—y)+yz{y—z)+zx{z—x). 19. a'^{b+c)+b%c+a)+c\a+b)+2abc. 20. (a+6+c)3-(a+6-c)3-(6+c-o)3-(c+a-6)3 21. (x-y)3+(2/-2)3+(3-a;)3. 22. a(6-|-c)2+6(c+a)2-|-c(a+6)2-4a6c. 23. ab{a^-b^)+bc{b^-c^)+ca(c^-a^). 24. a2(6*-c«)-|-6V-a*)+c^(a*-6*). 358 ALGEBRA Simplify : 25. ^(y+g) , y(8+a^) , 2(a:4-y). {x-y){z-x) (y—z){x-y) {z-x){y-z)' 26. ^ + y" + !! (a:-y)(x-z) {y-z)(y-z) iz-z){z—y) 27. ?^ + ^ + "^ (c-a)(c-6) ' {a-b){a-c)^ {b-c){b-a) 28. --4- + --J - + 29. bc{a—b){c—a) ca{b—c)(a—b) ab(c—a){b—c) b^—ac c^—ba a^—cb (a-b){b-c) "^ (b-c)(c-a) "^ {c-a){a—b) ' 30 bc(b+c) cajc+a) ab{a-\-b) (a—b)(a—c) {b—c){b—a) (c—a){c—b) 31. . , .:.■ ^. + ... 1^ .. +; 32. (x— j/)(z— a;) («/— 2)(a;— t/) (z— a;)(?/— z) (a-6)''+(&-c)^+(c-a)' a(62_c2)+6(c2-a2)+c(o2-62) * 33. SimpUfy (a+b+c)'^-{b+c)^-(c+af-(a+bf+a^+b^+c', being given that a is a factor of it. 34. Sliow that a—b is a factor of o"(6— c)+6"(c— a)+c"(a— 6). What may be inferred regarding other factors ? 35 . An expression is symmetrical in x, y and z and each term is ol two dimensions. When x=t/=z= 1, the expression equals 15, and when x=l, y=2, z=3, it equals 64. Find the expression. 36. Point out wherein it is obviously impossible for the following statements to be true : • (1) (a*+6*+c2)(a+6+c)=o3+6'+o2(64-c)+62(c+a)- (2) a^+b^+c'-Sabc=(a+b+c){a^+b'^+c^-3ab). (3) (a-6)(6-c)(c-o)=o6«+6«c+ca*-ac2_6c*-6a*. SUPPLEMENTARY THEOREMS 359 253. Identities. We have already had many examples of algebraic expressions which are identically equal, that is, which are equal for all values of the letters involved. Thus, (.T-f ?/)(x — 2/) = a;* — .V*, (o+6)3=a3+3a26 + 3a6* + ft', a3 + 6« + c3-3a6c = (a + 6+c)(o*4-62 + c2-a6-6c-ca). Any of these may be shown to be identities by performing the operations necessary to remove the brackets on one side, when the result is the same as the other side. Ex.— Show that (a+fe+c)^ =a3+&3^c3-3a6c+3(a+6+c)(a64-&c+ca). Here the cube of a-\-h-\-c may be found by multiphcation or by any other method. The brackets are then removed from the right and the terms collected. The two sides are now the same, which shows that the given statement is an identity. We might also have changed the second side into the first by factoring, thus : (a3 + 63-[-c3 — 3a6c) + 3(a + 6 + c)(a6 + 6c + ca), = (o + 6 + c)(a« + 62-j-c*-a6-6c-ca) + 3(a + 6 + c)(a6 + 6c + co), ==(a+6 + c)(o* + 6«+c* + 2a6 + 2ac+26c), :=-^(a + fe + c)^, which proves the proposition. 254. When two expressions are to be shown equal, the result may frequently be obtained by showing that their difference is zero. The diflference may be zero, (1) because all of the terms cancel, or (2) because it has a factor which is equal to zero, identically, or which is given equal to zero. .iGO ALGEBRA Ex. 1, — Prove (a-ft)3+(6-c)3+(c-a)3=3(a-6)(6-c)(c-a). Here we may prove that (a-6)3 + (6-c)3 + (c-a)'-3(a-6)(6-c)(c-o) = 0, (1) by removing the brackets when all the terms cancel, (2) by observing that (a — 6) + (6 — c) + (c — a) is a factor of the expression and this factor is identically equal to zero (art. 245). Ex. 2. — If a^h=^c, show that a'^-\-hc=h'^+ca. Here, as in the preceding, we may show that a* + 6c — t* — ca = 0. by showing that a-\-h — c is a factor of it and this factor is given equal to zero, or by substituting c = a-\-b in each side or in the difference. Solve this problem both ways. Ex. 3.— If a+6+c=0, show that {a-\-b)(b+c){c-\-a)-\-abc=0. For a-{-b substitute — c, for b-\-c substitute —a, and for c+a substitute —b and (a + b)(b-\-c){c + a) + abc = {-c){-a){~b'+abc = 0. Ex. 4. — If 2s=a-\-b-j-c, prove that s2+(s-a)2+(s-6)2-i-(5-c)2=a2+6Hc2. When the first side is simplified it = 4«» — 2«(a + 6 + c) + o»+6* + c*, = 4s«-2«(2s) + a* + i'* + c*, = o*+6* + c*, which was required. Of course, this could have been proven by substituting the value ol s at once. It is usually eeisier, however, to substitute in the last step. BXBRCISB les Prove the following identities : 1. a(6+c)2+6(c+a)24-c(a+6)2— 4a6c=(a+6)(6+c)(c+a). 2. {x+y)*+x*+y*=2{x'+xy-hy^)''. 3. (o+6)3+(a-6)3+6o(o+6)(o-6)=8a». StTPi'LEMENTABY THEOREMS 361 4. 2(a3+6'+c3-3a6c)=(a+6+c);(a-6)2+(6-c)2+(c-a)2|. 5. a(b-c)^4-b{c-a)^+cia-b)^={a-b){b-c){c-a){a+b+c). If a+6+c=0, show that : 6. (3a-26+4c)2-(2a-36+3c)2=0. 7. a2^62_c2^2a6=0 and c^— a6=62— ac. 8. (a+6)(6+c)+(6+c)(c+a)+(c+o)(a+6)=a6+6c+co. 9. a*+6*+c*=2a262+262c2+2c2o2. 10. (3a-6)3+(36-c)3+(3c-a)3=3(3a-6)(36-c)(3c-a). 11. a(62+6c+c2)+6(c2+ca+a2)+c(a2+a6+62)=o. 12. If a+6=l, prove that {a^-b^-)'^=^a^-]-b^-ab. 13. If a;+«=23, prove that -^ + -^ = 2. a;— z y— z 14. If a = ^-^—, b = , c — -, show that a-{-b+c-\-abc=0. X y z 15. If - H = r, prove that - + - = -. a a — c a—b a b c 16. If a; + - — 2/» show that x^ -\ — ^ = y^—2 x^ -\- — = y^ — 3y ; x*^-=y*-4:y^-\-2. X* If 2s=:a-\-b-\-c, show that : 17. s{s~a)+{s—b){s—c)=bc. 18. a(s-a)+6(s-6)+c(s-c)+252=2(a6+fic+ca). 19. (s-a)2+(s-6)2+(s-c)2+2(s-a)(s-6)+2(s-6)(s-c) + 2(s— c)(5— a)=5*. 20. (2a5+6c)(26s+ca)(2c5+a6)==(a+6)2(6+c)2(c+a)2 21. J_ + -L + -L_l= '^ s—a s—b s—c s s{s—a){s—b){s—c) 22. I6s(s-a){s~b){s-c) = 2bh^2c^a^+2c^a^-a*-b*-c\ 23. If6 + -=l5C + - = l, prove « + r = 1 and a6c = —1. 362 ALU EH HA 24.* n a 4- - = 3, find the value of a^ + 1 . 26. If a=a;(6+c), h=y[c-\-a), c=z{a-\-h), show that xy-\-yz-\-zx-^1xyz=^\. 26. If «4-y=o and xy=b^, find the values of x^+y^ and x'+y^ in terms of a and 6. 27. Eliminate x and ?/ from the equations x-{-y=a, xy=b^, 28. Eliminate x and y from z+2/=a, x^+y-=6^, x^+!/^=c'. 29. If x=a+6— c, y=6+c— a, z=c-{-a—b, show that xH2/3^z3-3x2/2=4(o3+fc^+c^-3a6c). EXERCISE 164 (Review of Chapter XXVI) 1. Show that x' + y' + z^ — 3xyz is divisible by x-\-y-\-z, and hence show that (fe-c)' + (c-a)3 + (o-6)» = 3(a-fc)(6-c)(c-a). 2. Prove that 3. If o + 6 + c + d = 0, prove that {a + b){a + c){a + d) = {b + c){b + d)(b + a). 4. Prove that (a — 6)" + (6 — c)" + (c — o)" is divisible by (o — 6)(6 — c)(c — a), when n is an odd integer. 6. If n is a positive integer prove that 12"— I is divisible by 11, 23»»+i-|.i by 24, 7«"-l by 48. 6.* Write down a quantity of the same typa is x^ + y' + z' — 3xyz of which ^x+}y — ^z is a factor. 7. Show that a, a — x and a — 2x are factors ot (a-6)(o-6-x)(o + 26-2x) + 6(6-x)(3c-26-2r>;}. 8. Show that {x + y)'' — x" — y" is always divisible by xy{x + y), when n is an odd integer. 9. If (^ — o)(l— a) = (^ — 6)(1— 6)^x, find x in terms of a and I only. SUPPLEMENTARY THEOREMS 363 10. If a; -f 2/ + 2 = 0, prove that (1) x* + xi/ + ?/^ = .y^ f 2/2 + z* = 2* + 2a; + x'. (2) (a;+2/-2)' + (3/ + 2-a;)3 + {z + x-2/)3 + 24x2/2 = 0. a b c 11. Simplify tc(„ — fc)(a _ c) + ca(6 - c)(6 - o) "^ ab{c - a)(c - 6)' 12. Solve (x-a)3 + (x-6)3 + (x-c)3 = 3(x-a)(a;-6)(x-c). 13. Show that (o + 6)5-a5-65 = 5a6(a-r6)(a* + a6 + 6*). 14. If 2s = o + 6 + c, show that (1) s(5 — 6) + (5 — a)(s — c) = ac. (2) s«+(5— o)(«— 6) + (5— 6)(s-c) + (s-c)(5-o) = o6 + 6c+ca. (3) (s-o)3 + (s-c)3 + 36(s-o){s-c) = 63. 15. Prove that a''(62_c2) + 6«(c2 — o*) + c"(a2-62) is divisible by (a—b){b — c){c — a) and find the quotient when w = 3. ifi «?• If X — a x — b x — c ' *i"^P*"y a{a - 6)(a - c) "^ 6(6 - c)(6 - a) "^ c(c - a){c - b) ' 17. If x = a^ — bc, y — h^ — ca, z = c* — ab, prove that ax + by-\-cz = {a + b-\-c)(x-\-y-{-z). a«(6 - c) + 63(c — a) + c3(o — 6) 18. Simplify -^r ' , ^ 'J, ^ .,. • ^ ^ (6 — c)3 + (c — a)* + (o — o)' 19. If a6 + 6c + co = 0, show that (1) (a + fe+c)« = o* + 6« + c*. (2) (a + 6+c)« = o»4-6' + c3-3o6c. (3) (o + 6 + c)* = a* + &* + c*-4a6c(o + 6+c). 20. Show that a;"+^— a;"— a;+ 1 is divisible by (x— 1)*, when n is a positive integer. 21. Write down the quotient on dividing X* — a* by X — a, x* + l by x*+l, o* — 32 by a — 2. 22. Factor x*-l-3(x«-l) + 4{x»-l). ,., o(fe* + 6c + c*) , ..,,.. 23. Simplifv -f Tv; — r + two similar fractions. ^ ^ (a— b){a — c) 24. Show that x(y*—z^) + y{z*—x*) + z{x* — y*) is not altered when V is changed to x-\-a, y to y-\-a, z to z+a. 25. If r*=«+l, show that x« = 5x+3. 364 ALGEBRA 26. Find two linear factors of (ax+h)^-\-{hx+cy + {cx\-a)3-~3(ax-\-b){hx-rC)(,cx+a). 27. If x^+y^ = z^, show that {x^-{-y^-z*)^ + 21x^y^z* = 0. 28. If a + b + c = 0, prove that a^ + b^ + c^ + 3(a + b)(b + c){c + a) = 0. 29. A homogeneous expression of two dimensions is symmetrical in X, y, z. Its value is 42 when x = y = z = 2 and is 16 when x=l, 2/= 2, 2=0. Find it. 30. Eliminate x and y from a; + 2/ = o, xy = b, x^-\-y^ = c. 31. If x+3/ = 3 and x^-\-y- = b, find the values of x^-\-y^ and x*+y*. 32. If o + 6+c=- 10 and a6 + 6c + ca = 31,find the values of a»+6»+c* Bnd a' + ft-'+c*— 3a6c. ANSWERS TO HIGH SCHOOL ALGEBRA ANSWERS No answers are given to elementary examples, oral examples or examples which may be verified or checked without difficulty. In each exercise the number of the first example to which the answer is given is marked with a star. Page 8 15. 108, 38, 10. 32, 60. 16. 3, 14, 39, 0. 17. 9, 29, 18. 19. 2. 21 . 25. 22. 44, 7. 23. 154, 616. Page 10 9. 47. 10. 70. 12. lOz+10. 13. xft.E. 14. 15». 16. 2a8+2a2+3a. 17. 11,5a;. Page 12 24. 37. 25. 17. 26. 34. 27. 1. 28. 1. 29. ^. Page 14 30. (1 4- A^ hours. 31 . (5a;+20y-72) cents. 32. ^±^ ceata. 33. 5x+10y+5(K. 34. 1234, 4019. 35. 3, 02. 2, 16, 05, 03. 36. - + -. 37. 20, 20. 38. 24. 39. 2*. X y S67 968 7. la+eb-15c. 10. 4o+46+4c. 13. 8o-4fe. 16. 2x+2y—z. ALGEBRA Page 38 8. lOx^-Ux+9. 11. 'ia-8b+3c-5d. 14. 6a2+862_6c2. 9. lOa-76. 12. 8a;— 62/+5«. 15. 3a4-36+3c+3jl 18. 0. 19. 15x+5y. 23. 2o— §6— ^. Page 40 20. 46c. 21. 10a2+o6. 22. 4.V*. 13. -3xy. 14. 0. 18. 0. 19. z. Page 41 15. 4p2. 16. 10TO-3n. 17. 6.V-4?. Page 44 15. 3a-2b. 16. 2a+5c. 17. -Sx^. 19. 4a2_4o-15. 20. 26. 21. 13r-p. 22. 36-oc-2a. 23. z^+Gar-S 24. 2a2+a-12. 25. a+b+c. 26. 2a;-3. 27. 10x»+2a;2+8x4-2. Page 46 11. 3x+2y. 12. -2a-36. 13. a+b. 14. 6-a. 15. 3a+6-3c. 16. 3x2-3. 17. 7. 18. 11. 22. 6, 4, 4, 6, 10. 23. 4a+46-15c, 4o-46+4c-4rf, y, 0. ANSWBBS am Page 47 1. o. 2. 4x46. 3. 5n. 4. 5c. 5. 46,8a. 6. 2a+26+6c. 7. — a;+y-5z. 8. 3a-2b—2c. 9. 14, 6. 10. -7. 11. y-x. 12. 31. 13. a-2c, 2c-o. 14. 9m-2n. 15. 4a;-9. 16. x-U. 17. 26-4c. 18. 56— 5a, 0+36— 4c, 7a— 6— 6c. 19. •2x. 20. 3a;— 6. 21. o-^6+^c. 22. 52-3a:. 25. 20. 26. l + 2x. 27. 7. 38. 7n+4a;-2TO. 29. 3-o-6-c. 30. 5c-36. Page 52 19. 1, 4, 5, -3, -1, -8, -9, 7. 20. 3, 16, 35. 21. a2, -o3, -8, -1, 1, 81, 32. 22. 29, 81. 23. 24. 25. 90. 26. 6. 27. 30. 28. 23. 29. -20. 30. -50. 31. -100. Page 53 13. 8o+76+9c. 14. x-4y. 15. dm. 16. 9o-6. 17. 4o+i6. 18. 6x2+8x. 19. a^ 20. x^-9x^+10x. 21. -4a6. 27. 6x2- 15a;. 28. 7a*-5a. 29. 2as/, 3x^+xy+Zy^, x^+5xy+y^. Page 56 19. 2af2+4x-4. 20. 5a2-8a-22. 23. 214. 24. 'ki^-db\ 25. 2x^+2y^, ^y. 26. a^+ab+4ib^. 27. 14a;+30. 28. x2-6x-7. 29. 2x-10. 31. 12x2+12. 32. 3x2+10. 33. 15a. 34. 3x2+12x+14. B B i70 ALOEBHA Page 60 18. 1. 19. x^-Zx+2. 20. 2y. 21. 5. 22. o-b. 23. a\ 24. x+13. Page 60 2. 9, 16, -12, 25, -7, 27, -64, 91. 3. 1, -1, 1, 16, -27. 4. 4a2. 5. 8a2-9a. 6. Sa^-Sft^. 8. 30a+406. 9. 12x2+12. 10. 4x2+12xi/-9i/2. 13. 13m2+137i2_24TOn. 16. 4x2. 18, 5a2_3ai_452. '20. 3a2-12a-rl4. 21. 6x2-2x!/-6!/2. 22. 4-a. 23. 8, 19. 24. x*-16. 25. 8o2-9a-l, 6-lOa, 3a-4. 26. a^-b\2,a^-b\ 21. 20b^-obc. 28. 0. Page 81 22. 7, -2. 23. -8. 24. 5, -2. 25. 5, 6. 26. ^, 2. 27. 6. 1. Page 83 21 . 4, 5. 22. -3, -3. 23. 4, 9. 24. 5, 3. 25. 12, 12. 26. 19, 3 27. 15, -56. Page 92 28. 2a. 29. 1. 30. x+1. 31. 3a;-8. 32. x+5, a+b. 33. 2(x-2)(x-3). 34. 3(a+4)(a-3). 35. a:(x-7)(a;-l). 36. ±5, ±1. Page 93 17. 7,-11. 18. 10,-4. 19. 5. 20. 10 in. 21. 7 in. 22. 14 in. 23. 3^ in. ANSWERS 371 Page 95 17. 2a;»+2. 18. 2a^+2b^. 19. 5ar«+5. 20. 4tah. 21. 5OT*-10TOn. 22. 5z^+24:xy-5y^. 23. 3x*+12a;+14. 24. a;2~4. 25. 18a- 15. 26. 3x2-4a;2/4-6i/2. 27. 16z-34. 28. 36a;. 29. 9a^-8ab+%^. 31. 8. Page 98 25. 5o«-5. 26. 3a2-862. 27. 0. 28. 19q'-4:pq. 29. x<-a«. 30. 15. 31. S{x+y){x-y). 32. 5(x+2)(x-2). 33. a(o4-l)(o-l). 34. m(x-a)(a;+a). 35. 5(l + 3p)(l-3p). 36. {x+y){x-y){x^+y^). 37. ;7(ii;+r)(i?-r). 38. {x+l){x-l){a+b). 40. a2-2a6-362. 41. 2a;, 7. 42. 4, ±8. Page 102 22. a;(x+l)(a;-l), 3(x-2)(x4-2), a(a-l)(a-2). 23. 8,-2 ; 2, -1. 24. 2. 26. 2(x-2)(a;+2)(a;2+4), (a+2)(a-2)(a+3)(a-3), 2m(m+3)(m-3), {x+y){x-y){a+b){a—b). 27. 262+2c2^4. 28. 3x2-5i/2. 30. 12a6-3862. 31. 43,23,17,13. 32. Sx+6y. Page 104 13. a + 2b. 14. a— 6. 15. tw— n. 16. x+y. 17. m + 2. 18. a-2. 19. x-3. 20. j/-1. 21. a+b. 22. x-5. 23. 2(3a+26). 24. a(a-l). 25. 2, 3. Page 106 22.^+1. 23. -J^. 24.^:^. 25.^^. 26.-^. X y—^ X— 3 m a-f46 BB 2 27. ^— ^. 28. o+l. 29. ^^1-^ 30. x^^l x+2y ^ 2 372 ALOE BRA Page 107 13.^ ay x-\-2 a— 5 10.?. 11.4. 12.-. 13.^ 14.^^. IB. 1 Page 108 10. o«(a + l). 11- 3x(x+2). 12. ab{b+c). 13. 2(a;«-l). 14. ar(x+j/)2. 15. (x4-l)(a;-l)(2;-2). 16. aA(a-6). 17. (a+6)(a-6)2. 18. x(x-l)(x+l). 19. 4a:(x- l)(x+l). 20. (t/_l)(j/+l)(y_2). Page 109 22. ?^±^. 23. ^^-^y . 24. ^±^. 25. ?-t^. 26. 0. 6 8 12 x^-y^ 21.^. 28. -"^ . 29.^-^. 30 ^+^^ X 6(x+2) ah(a—h\ (o+l)(a4-2)(o+3) 2 31. — (a-l)(a+2) Page 111 2. X— 2,60(x— 2). 3. x+y,xy{x+y). 4. x-2, (x— 2)(x+4)(x— 5). _ a — 6 x3/i6 e^ .yj a X x4-2 2(x— 3) a+& a a 2a 9 13. 18. 6-4c 46c x-2 x-3 46'c "• '• •■^- ' " x-3' X4-3' 3(x-2)' a-h ^^ a— 2x X— 1 a2 'x-2 11. !|,1. 12.0. 14. ^ ,"" " . 15. 0. 15 xz '«• A- "• Zv 19. 1. 20. {a^b)\ 21. 3. 22. +4. ANSWERS 373 Page 116 18. 7x+6. 19. 0+6. 20. 3~x—y. 21. 3z+3^+3z. 22. 22x2 ^24x-ll. 23. 10x2-5a;+15. 24. a;2+8x— 12. 29. a. 30. 2. 31. 4|f. Page 119 17. 3a;2-12x+ll. 18. 0. 19. 2ad+2bc. 20. 0. 21. 12x2 + 12. 22. l-x8. 23. x*-10x3+35x2-50x+24. 24. x*-10x2+9. 25. a6-l. 26. 13. 27. 0. 28. 0. -^ 29. l + 3x+6x2+10x3. 30. 2x3+9x2+3x-l. 31. 28x^+x*y-S3x^ij^+3lx^y^+20xy*-12y\ 33. 2-x+4x2-2x3 34. 195. 35. 5i. 36. abx*+x'^ib^-ac)+adx^+x{bd-c^)+dc. 37. p^x^+x{pr-q^)+qr, x3(a2_a)+x2(a2+a_i)_i. 38. 2a^y^-2b^y+2bc. 39. px^+x{p^+Zp+3). Page 123 30. x2+2x+l. 31. -x3+9x2-l. 32. a^+a+1, a^-a+1. 33. -2xy. 34. 2a. 35. 6. 36. a-2. 37. a^+Sa— 2. 38. x^+xy+y^. 39. 2aa;. 40. x+c. 41. a:+p— 1. 42. ax—b—c. 43. ay+a+l. Page 125 1. 4. 2. -5. 3. 22. 4. 2/+5. 5. 2j/'. 6. -2y\ 7.\ + -l-. 8.1+—. 9.2--^. 10. 5x-3+~^. x+l a—b a-\-b x+2 1■^. l+x+x2+x3. 12. 1—x+x^—x^ 13. l + 2x+2x2+2x3 14. l+2a+3o»+o'. 15. o-3. 16. 6. 17. z-i, 7, 374 ALGEBRA Page 126 15. a3+63+c3— 3aAc. 16. a»- 256. 17. x2-4x+8. 19. 21. 20. llx2-7x-8. 22. x^{c-a) + x{d-b) + (f-c). 23. 7. 26. 5!/«. 27. a;+6i/-22. 28. 2(a6+6c+ca). 29. bx^-bcx^^x{ac-a+b)—bc. 30. 35. 31. x*— 4a;2+12x. 32. 3x-8. 33. 3*. 34. 9. 35. G. 36. p^+p-2. 37. -52. 38. 4. 40. x*+2x^+3x^+2x+l. 41. 9. 45. x*+x^(b+p)+x^q+bp+c)+x(bq+pc)+cq. 46. 3a+2b—c. 47. x*+x^+x^a-a^)+x(l-2a)-l. 48. a^+2a^bc+iabh2+Sb^c^. 49. a;3— x2(a-)-6+c)+x(a6+6c+ca)— aic. 50. 6— c. 51. 3a:5_10a;* + 3x3-14a;2-7x. 52. x^+y^+xy-2x-'iy+i. Page 129 21. 3(x-2)(x2+l), a(x-l){y-l). 22. a+b, x-1. 23. (2x-yX5x-32), aJ(a+c)(a-36). 24. x-3. 25. (6x-7y)(8o4-56). 26. {x+y){x+y-\-A), {a-b){2a-2h-l). Page 131 34. 14x2+191/2. 35. 3a2+362+3c2. 36. 2x«+6x2+2. 37. 3a2-f362+3c2-2a6-2ac-26c. 38. 8{x^-z^-xy+yz). 45. 3(x+l)2, a(a+26)2. 46. (a+6+2c)2, (a+b-c-d)^. 48. 3. 49. 14. 50. (x2+2/2)(a2+62+c2). 51. {ax+by)^+{ay-bx)\ 52. 0. Page 134 11. a2-62-c2+26c. 12. 4x2+ 12xy+9j/ 2—25. 13. p^-iq*-9r^+l2qr. 14. l+x2+x* 15. a*-62+c2-d2_2ac-26d. 16. a^+Ab^-c^-4:d^-4dab+'icd. 44. 2(x+2)(x-2), o(o+l)(a-l), (a-x)(a+x)(a2+x2). 46. 6{o-6+2c)(o-6-2c), (x-3b){x-b}{x-5b). AN S WEBS 376 Page 134 {continued) 46. {b+c){b—c){a+d){a—d), (a+b+c)(a+b—c){a—b+c){a--b—c). %7. {ax-\-c+by){ax-'t-c—by), {7n—n+Smn){m—n—Smn)f 48. (a;+l)(x-l)(3a;-2), x{x-l)(x-3)(x+3). 49. 2a2— 2a6+26c— 2c2. 50. {x+y){x—y)(x+y+a){x+y—a). 51. 2a2_6a-fl, 12xz—24yz, 24o+9a2— Ga', 20a;2|/2— 40a;3?/. 52. a«+6*+c*-2a262-262c2_2c2a2. 53. (x-y){y-z)(z-x). 54. (a-6)(c-a). Page 137 13. 2(x2+2x+2)(x2-2a;+2), x{x^+x+l)(x^^x+l). 14. {a-b)(a+b)(3a-b)(3a+b). 15. (a;2-a;+l)(x2+2;+l)(a;*-a;2+l). 16. (a+6+c)(a+6-c)(a-6+c)(a-6-c). 17. (a2+3)(3a2+l). Page 139 31. 3(a;+8)(x-9). 32. 2(a+l)(a+3). 33. a:(3a;-l)(2a;-l). ^4. (x+l)(a:-l)(x+2)(a;-2). 35. a(a-l)(a+l)(a-3)(a + 3). 36. (a + l)(a-l)(3a+l)(3a-l). 37. (a;+l)(x+3){a;-l)(a;+5). 38. (x-2)(x-7)(a;4-l)(x-10). 40. a;2-5a;+6. 41. 4a;2-16a;+15. 42. ±1, ±11, ±19, ±41. 43. 33a^-38ab-Sb\ 44. (a;+2/)(a:+42/+l). 45. (3a+26)(a-6+2). 46. x^+l. Page 141 22. 2(a-2)(o2-|-2a+4). 23. 3(y+3)(y^-3y+9). 24. a{a+l){a^-a+l). 25. 6(a4-6)(a2-a6+62). 26. {a^+b^){a*-a^b^+b*). 27. (a;+2/+a)(x2+2x2/+2/2-aa:-ai/+o2). 28. x(x2-6x+12). 29. {2a-b){a^-ab+b^). 30. 2o(a2+362). 31. x+y. 33. o2(a*-6a26c+1262c2), 2/22(3a._2^2)(9a.2^3a;y2+t/222). 34. (a-6)(a+i)(a2+62)(o2_o5^t2)(a2_,_(^^52)(„4_o2ft2^ft4). 35. (x+l)(x-2) 36. 2. 376 ALGEBRA 7. (x-l)(2a;2-9x-4). 9. (x-l)(x-2)(x+3). 11. (a-l)(o-2)(a+4). 17. (a-6)(a+26Ka.+ 36). Page 143 8. {x-\){x-^\){x-2). 10. (a;-2)(x-3)(x-f5). 12. ''?^/')2fa-26;. 16. -12. 20. 8. -4 21 . 2, 3. Page 147 4. 2a;2+2a2-262. 5. 4x3. 7. 14a2 + 1462+14c2+14a6-10'?/:-22ftc. 9. 14,860,000. 10. -5. 31. 3a(2a-6)(4a2+2a6+6-; 33. (3x-4)(4x+5). 35. (x+2/)(x-j/+l). 37. {x—y){x^—xy-\ry^). 39. (a— 64-c)(a— 6— c). 42. (a+26)(a-26-3). 44. (a+&)(a+6+c). 46. (x-j/)(x2+xy+j/2+x+2/4-iy 48. (2a+56)(2a-56 + l). 50. (x2+4x2/-2/2)(x2-4x?/-!/2). 52. (x-l)(x2-10x-3). 54. (x4-l)(x-l)(c+l)(c2-c^l;. 55. (a + l)(a-l)(a+2)(a2+l)(a'*-2tt^ 1, 40. (x+2/)3. B. 4a2+462+4c^ 8. 19997. !1- 2a+19'7. 30. (x-2)(4x-9). 32. 8(a+c)(c— a— 6). 34. 4(3a-5)(9a2+15a+25). 36. (x-3)(x+3)(x2+2). 38. {x+Uy){x-\2y). 41. (x-36)(ax-2). 43. {2x-y){2x+y+a). 45. (o-6)(a-6-l). 47. a6(a+6)(o-6)2. 49. 96(4a2+2a6 + 62) 51. (a2-62+a-3)(a2-62-a + 3). 53. (a-l)(3a2_2a-10). 60. (a-6)(6-c)(c-a). 61. (x-2)(2x+3)(3x-2). 62. (x-l)(x-2)(x-3Kx-4). (x4-lHx+3)(x-2Vx-6). 63. 1, 5, —6; 0, 1, 6, -7. 65. x^—c\ 66. o^-ftM c^+2ac. 67. —4, 5. 68. (x-a)(2x+a+6). 69. (oi+cd)2-(ac+M)2, {ab-cdf-{ac-U)'^. ANSWERS 377 Page 153 16. 30. 17. 13. 18. 1,3,10. 19. 29. 20. 1,3.5. 21. 16. Page 155 6. 45. 7. 4. 8. 1. Page 169 6. About 2 h. 35 m. after A started ; 31 m. from Toronto. 7. (a) At 10.55, 2 m. from C towards D. (6) 22 m., 17 m. (c) 11.10. Page 172 5. 13. 6. A square, 16. 7. Right-angled, 4. 8. M. 9. 54. 10. 16; 5, 6i, 8. 11. 6. Page 177 3. 5, 10, 13. 4. 6|. 5. 13. 6. 7|, 4. 7. 30, 30. 8. (1, -7), (-3, -17), (5, 3). 9. 112i. 10. 24. 17. 24. 18.(4,4). ' 25. (3, 2), 90°. 26. S1200, 12th. Page 181 I. 4xyh, 24ax^y*z': 2. x—y,xy{x^—y^). 3. a-\-b,b{a—b){a+b)^. 4. x-Z, (a;-3)(a;-4)(a;+3)(a;+5). 5. a+5, (a+5)(o+3)(a-7)(a-2). 6. 3(a:-2), 3(a:+l)(a;+2)(a;-2)2. 7. x-y, y{x-y){x+z), 8. m-2, 4m^2(^_^2)(m-2)2(m2+2m+4). 9. 2(a»+o6+62), ^{a^-b^). 10. a+6-c, a(a+6-c)(a+6+c). II. o+ft+c, (a+6-f-c)(a— 6-c)(6-c— a)(c— a— 6). J78 ALGEBRA Page 181 [continued) 12. r^-3-.v + y«, (r-f yKx^+aV-rJ'';- 13. 3z-2, (3a;-2){a;+3)(a:-3)(2a;-3). 14. 5x-l, (5x-l)2(5z+l)(2o+3c). 15. a;-3, a:(a;-3)(x-2)(a;«+5). 16. u — v, {u—v){u+v)(u^-\-v^){u^-{-uv-\-v^). 17. x2-8, (a;2_8)(a;4-2)(a;4-3). 19. -a. 20. 2;2_3a;y_^2?/2, x^+xy-B?/^. Page 182 1. x-l,(x-l)(x-2)(x2_5x4-3). 2. a-l,(a-l)(a-5)(o''-18o-l). 3. x-2, (x-2)(x24-4)(2x2-3x-6). 4. a-1. (a-l)(a2+l)(3a2+a+6). 5. x-l,x(x-l)(x-f4)(x2+x-6). 6. (x-2)(x2+5x+l)(x2-2x-l). 7. , °7^, , . -1—. 8. x3-6x2+llx-6, x3-9x2+26x-24. a^+2ab-\5b^ 2x4-4 Page 186 1. x-5. 2. (o-3)(a-4). 3. 2(3x2+2x+2). 4. 2x-9. 5. 262-6-5. 6. 2x-ly. 7. a-2. 8. x-3. 9. 3a2(o-l). 10. x-1. 11. (x-3)(x+l)(x+2){x2-x+l). 12. (x+l)(x+2)(x+3)(x+4). 13. {2x+3)(3x-4)(x2-f3x-l). 14. (x-l)(x-2)(x-3)(x-4). 15. (5x2-l)2(4x2+l)(5a;2+x+l). 16. 3. 17. 35. 18. x2^5x-14. 19. 11. Page 187 1. x-U, (x-9)(x-10)(x-ll)(x-13). 2. x-3, (x-3)(x-12)(x2-2)(x24-3x+9). 3. a-b, (a-6)2(o+6)(o-+a6-f 62), 4. x-f 3, x(x+3)(x+2)(x-4)(x-5). ^. (2a+l)(o-3), (2a+l)(a-3)(a+3)(2o-l), ANSWERS 3T9 Page 187 {continued) 6. x-b, {x-a){x-b){x-c). 7. a;-l, (a:-l)(a;-2)(a;-3)(x+2)(a;+3). 8. (x-l)(a;+3), (a;-l)(a;+3)(a;2+a;+4)(a;2-6a;-4). 9. (a-r3)(2a+l), (a-2)(2a+l)(a+2)2(a+3)2. 10. {x-y)\ {x-yf{x-2yf{x-\-2y)-'. 11. a;2_a;y4-2/2, (a;2_a;2/+2/2)2(x'+a:2/+2/2). 13. 3. 14. a;*-a:2a2+o*. 17. 1, 3. 18. a;2-3a;+2, a;2-6x+5. 3a;+2 19. 2a;+3 Page 191 ""^^ 8.-^. 9. °~^ • 10. ="+2 o2+o+l a;-2 4o2+3a-6 23?{x+\) 11 ^^ 12 =^'-^ 13 °' -3 14 ^-3 ' x+1 ' l2x^-lx-4: ' a«-2o3+2a-5 " 2a;- 1" Page 192 ^ a'+b\ g_ 2y 3.-1^. 4. ® o«— 6* a;2-^2 a;2-y2 o«-7a+10 5. J^. 6.-^^. 7.^^. 8. ^ . 9.0. a«-6* 1-a* a;*-?/^ (a;+T/)(2z— y) 10. 3 ^^ 3x^-5ry-2y\ ^^ 5^ (x+^){x+5){x+l) x2-i/2 a;2-5a;+6 13. -iL.. 14. ^„. ^5.^±^ 16.0. 17, 1 x—y a^—b^ x—y x^—Y 18. — ?— 19. ? 20. 0. 21. 2. 2o-36 (a;-l)(a;-2)(x-3) 22. 0. 23. 1. 24. — ^— . 25 ^^ ^^ 2(a+l) x'^-y^' x*-2/* 3 ^?L 29 -^ a4i_y8 ■• 81-x* ' c»-62" ■ l_a;8 26. -M,. 27. ^*^. 28. 44. 29. -^ . 30. 2. 8W ALGEBRA Page 196 1. ^~° 2. ^ 3. J?_. 4. ~^ oa;(a+a:) 2a-36 x-2 x^-V g 26+3a g x»+ax _ ^ a;'-a:+2 _ g _1_ ab{a—h) ' a{x—a) ' x^—1 ' x—2 9. -^.. 10.0. 11.^. 12. „^. 13. 20 x^-y^ b-3a 3a+2x, x*-l 14. °'+^' ■ 15.: V ,• 16. ac{a—b) (c— o)(c— 6) (x— o)(a;— 6) 17. _£±y_. 18. _1— . 19. 0. 20. 0. 21. 0. x{x-y)» 1-9x2 22. 1. 23. " ""^ ^" -.^y-y^-^ _ 24. -1. 25. 0. (x— y)(y-2)(z— z) 26. d. 27. , .^\ . . 28 ^^"^ {a+b){a^+b-^) (x»-9)(x»-25) 29. —^- 30. ^^ (o-fl)(o+2)(a+3)(a+4) (a;«-l)(x»-9) Page 198 1. 1 2. -J—. 3. ^I^. 4 ?!±i'. 5 -^. 6 y* o»— 6* ' x+4 ' o ' o»— 6* ■ x*+y« 7. x*+l-f-l. 8. o*+l. 9. 1. 10. 1. 11. '^. X* a* X— 3 14. ? + ??+l. 15. ^l-l + g. 12. y(°~^) 13. a+&— c ANSWERS 381 Page 198 (continued) 16. --^-+J!-*. 17. <^. 18.?!^Z^». 19.i±2. x^+y^ {a — 2)^ a^ x—2 20. -±-. 21. ^. 22. 1. 23. J-. 24. ^(°+f ^ . o*— x2 o2 a— 8 a— 66 Page 200 1. _?_. 2. Z^. 3. -i-. 4. ^. 5. 1. 6. -?. 106c 56 0^—62 a:i/ X 6 7. -1^. 8. 1. 9. 1. 10. ^y+\ . 11. i'. 12.^-^;. 13.?. 14. a+6. 15. a+6. 16. ^,. 17. ^, Page 201 1. -i^. 2. «?^^ 3.-?^. 4.0. 5. -?-. 6.2. ___^a*— 6* 6c --» a;2— 9i/2 ^ l—a* x 7.1. 8. 1±^'. 9. ^— ^ 10. ^+^ 11.-^+^ 2 2a; 6+c x^+iar.+S o«_oi+68 12. ?!4\ 13. 5°+-) ■ 14. i±^. 15.0. 16.1. 0+6 a*+ax+x2 ^ ,..« 17. ,„ ^"^"^^ ^ • 18. -i-. 20.x. 21.1. 22.1. (2x+3)(3x-2) x-1 25. x+x^ 26. ?. 27. -1. 32. "^ ■ 33. *' 6 a+6 (a+b){a^+b') 34. 0. 35. 4^^:10x gg Q ^__ («-l)(a:-2)(x-3)(x-4) x+j/ 38S ALOBBRA Page 210 13. — — -. 14. -. 15. a—h. 16. a-\-h—c—d a-\-h a-\-h—c 17 ^^. 18 "''+°^+^^ 19 ^ 20 _^±^ 21 - ' a+6 ■ a+b ' ' 2 ' " 2 ' ' 6* 22. ^'. 23. a-b. 24. ?^. 25. -^, ^. a2 2 6+c 64-c 26 ""+^ om— 6 __ a6 -g 3a6— 3 a^ _g wn(a+fe) TO+n' m-{-n a—h ' o+3 , * mn—m—n o/\ oft— cd -^ 2s 2s— In 28— an 30. — — -. 31. — — , , a-f-b—c—d a+l n n 32. 8-,r-\-rl, ^-"+° li:^. 33. ?^=^, H^=^. r a— i 2t fi Page 213 13. a, -h. 14. ^i:^, ^illjl. 15. 2a, -b. 16. o, 6 a, — Oj Cj — 0, ^^^ oA-oA^ a^V-oA . 18. b+a, b-a. 19. ^a. ife. "2^1 — O1C2 O1C2 — fflaCj 22. c, 0, a. 23. 4a -36. Page 214 25. °° ^'f . 26. afcc. 27. 5,5. 28. 21Jl, 27A. a — 36 29. 2,-^. 30. J, 2. 31. -3. 32. $543, $457. 33. |§. 36. $16400, $13600. 37. 1540, 880, 616. 38. ??i::!^^, ^±Z^. 39. 2a='+26^ 40. 35, 2n n^—n 41. $2100, $560. 42. 182040. ANSWERS 383 Page 219 18. a^-fSa+l. 19. 6. 20. -4. 21. {z+l)(x+2)(x+3). 22. l-a:-2T2, 2-3x-|a;2. Page 221 19. ix+y)^-2{x+y)-\-l. 20. x^-3ax+a^. 21. 024.6?. 22. a2+62. 23. x^ + 2 + - . 24. -6. 26. 13. Page 223 21. 2a^+{iab\ 6a^b + 2b^ 23. 2a^+2b^+6a^b+Qab^+6ac^+6bc\ 27. 27. 28. 242. Page 225 M. x'+x+l. 18. l-2a;+3a;2. 19. ?_1J.?. 3 x 20. 3a2-4a4-l- 21. l-a;^. 22. 4c. 23. x-l. 24. a-3. 25. x—2. Page 226 1. 3z2_4xy+22/2. 2. x3+2x2-3x+l. 3. x«+3a;*-2a;2+2. 4. hx^-lx+l. 5. 5a;2-2ax-3a2. 6. 2x2 + 3a+7. 7. (a:+2)(x+3)(a;+4). 8. (a;+l)(x-5)(2x-3). 9. 2x2-5a:+2-?. a; 10. 3-5a;. 11. 2x^-x-^l. 12. a. 13. l-a;-^a;2, \-^a-la^ 2+^x-^x^ 15. 7a;2-2a;-3. 18. 8a3. 21. 0, -81/3. 24. 2x^-3x^+x-2. 25. 16. 27. 6x-4. 28. 7a;*-2ar+l. 984 ALOEBRA Page 230 32. Vl3. 34. V2\. 35. 2\/2, 4\/2. 38. 5\/2. 37. 4\/2, 12\/2. Page 232 9. 10\/3. 10. 7\/2. 11. 5\/5. 12. -3V7. 13. 7^2. 14. 8VTT. 15. 7a/5. 16. -4\/2. 17. 866. 18. 7*94. 19. 11-62. 20. 5-20. 21. -141. 22. 25-46. 23. ±6-083. 24. ±3-873. 25. ±6-782. 26. ±6-481 27. ±9-592. 28. ±13711. 29. 7483. Page 234 13. 24\/3. 14. 12\/7. 15. 5+2V6. 16. 27-4\/36. 17, 30 + 12\/6. 18. a+6+2\/^ 19. 2+3\/2. 20. 12+V6. 21. 6 + \/l0. 22. 6a+66-13\/^. 23. 6+2\/l5. 24. 4V6-4. 25. a+6-6-\/^. 26. 1. 27. 6+2\/3+2\/2+2V6. 28. 16+4Vl0-2Vi5-4\/6. 29. 2a-\-2Va^^'^. 30. nx-5y-\2^^^y\ 31. 1. 32. 12-W2. 33. 6V6. 34. 70. 35. 30-5\/6'. 36. V8+\/7. 37. 42, 43. 38. 46. 39. 9\/2. 40. 30-92. Page 236 13. 14+8\/3. 14. 6V2+W3. 15. 5+2^6. 16. °~_^ ■ 17. Vl5. 18. ^^~^^ . 19. -577. 20. 3-636. 21. -817. ANSWERS 386 Page 236 {continued) 22. -318. 23. 1-491. 24. 084. 25. 1-225. 26. -894. 27. -072. 28. 212. 29. -82. 30. 1-39. 31. 3-15. 32. 11-71. 33. ±2-73. 34. 1-008. 35. V2. Page 238 1. 12a/2. 2. 12\/6. 3. 10V3. 4. 62. 5. 191. 6. -4^2. 7. 22-12^2. 8. 12-W6-2VZ+W2. 9. ^. 10. 9-4-V/5. 11. 1. 12. 2VT3+2V2. 13. 74+11V6. 14. I. 15. 1-732. 16. Vl2+Vl0. 17. 1, 2. 18. 2V2, ^V6, iV30, ^VU, i(4v'2-2\/3). 25. ±8-661, ±7-937, ±9-899, 1-291, -518. 26. -817, -447, -414, -757, -337. 27. 25V3. 28. 2\/2. 29. lilZ^, 4\/6. 30. 202. 31. 30. 32. 6. 33. 4-83. Page 241 1. a;2+a;-132=0. 2. a;2-a;-156=0. 3. a;2-49=0. 4. a;2+6a;- 112=0. 5. 5a;2- 6a; -440=0. 6. x2+6x-9400=0. 7. a;*-19x+88=0. Page 242 1. 6, -1, -22. 2. 6, -25, 21. 3. 8, 19, -15. 4. It, -11,2. 5. 1, -10, 9. 6. 2, -5, 2. 7. 1, 4, -32. 8. 6, -27, 28. 9. 9. -19. 44. 10. 2, -6, -3. 11. 0. 2, 7. 12. 0, 1, -1. 386 ALGEBRA Page 2^ I. 4-236 --236. 2. 7-828, 2172. 3. 1-646, -3-646. 4. 1-916, ^9-916. 5. -232, -3-232. 6. -851, -2-351. 7. 3±Vll. 8. -4±3\/3. 9. i±V2. 10. l±^vTl. II. 5_±V157^ ,2. -J±iV2-2. o Page 251 20. 1-618. 21. 2^, -1. 22. 5, 12. 23. 6, -2. 24. 14, 6 or 16, 4. 25. a; = l or 2, t/=2 or 1. 26. 540. 27. 3-236, -1-236. 28. 20c. 29. 60, 90. 30. 1-449, --949. 31. 20. 32. 3 m. perhr. 33. 8. 34. ^, , — a—b a+b 35. 20, 30. 36. x=2 or ^. 37. 4. Page 255 18. 2-54, 1-0936 19. 8 : 27. 20. -192, 1-302, 21. 3937 : 6336. 22. 4:5, 11 : 27, a+3 : o+5. 23. 11:15,13:18,2:3.3:6. 31 . «-^^ 32. 1±^ . c—d l+4a 33. 4:6. 34. — -^ • 35. 10. 39. 20fem : an. b+c b+c Page 259 29. 11-55. 30. 10|, ^. 31. AE=6l DE=1\. 32. 240. 33. 9-899. 34. 2 : 3. 35. 2, , -^— , - ^ a — c m — n p ANSWERS 387 Page 259 (continued) 36. ior§, 5or -1. 37.- = -= — . ^ ^ 5 3-8 38. AC=20, AE=5, DE=i. 39. 147 ft. 40. 3 or ^. 43.5:4:2. 44. ^ or -i. 45. 61, 5J ; -^,. -^ . b-{-c b+c 46. 2 : 3. 47. 17^, 25, 30. 48. 110 : 15 : 17. Page 262 6. 6 ; 7^ ; U^, 5, 4,% ; 5, 2f, 1*. 8. 2, 8. 9. 3, 6, 12. Page 264 4. H- 5. -17^. 6. Ih 14. '^. IB. V- 17. -^. 18. 7: 16. Page 266 17.-^. 18.^. 19. 2, |. 20. f. 21. I, |. 22.6. 23. 1 = 1 = 5. 24. ft. 25.^,-1. 26. i. 32. ±10, ±5, +6. 38. 41|. 39. 1,3,4. Page 270 1. 3a, -a. 2. b, -5b. 3. Zm±mV6. 4. - 2p±^\/6. 5. o±V5236. 6. -6±a/P+c. 7. -l±^--f 1. g^ --b±V¥^ac^ Q fe±A/6''+4ac .^^ ?±vg^^^ a '2a ' 2p CO 2 M8 ALGEBRA Page 271 2. I, I. 3. ^, -5»5. 4. i±\Vn. 6. i,3±^V89. 7. A. -^3. 8. .^. -j\j. 9. 2a, -36. 10. 7, J^i. 11. i, -\. 12. :j>5, -jig. 19. ,%±Ti5\/l6l. 20. 3±Jv/57. 21. ^, -^. 22. ^±2\/5. 33. ±V6. 35. l±2\/6. 38. 2'414, --414. 46. 2-786 or -120. 1. 1. §. 5. A, -T%. 32. 1±2V2 39. 3. 43. 6-18, 382. 63. 7-03, 8-78, 8, 2-29, 6-42, impossible, Page 276 13. 1, -6, -|±^V^39. 14. 6, -3, %±\y/^^l\. 15. 2, -l±\/33. 16. ±2, ±2\/^. 17. 3, 2, -5. 18. 1, I, |. 19. -|±^V^=23. 20. 3, \, -i±i\/^, -ft^v'^a. 21. 2, 3, -1, -2. 22. 1. 1, -l±\y/l. Page 277 25. '*^^, !ii^- 26. -4,3, -i±^V^^. 27. 5, -f±|A/=^. 29. a+h±V'^^db+b\ 30. -, --• 31. 15. 32. —, — . c a o— c o+c 33. -1, -2, -4, -8 36. a+6. 0, «'+^* 34. 12. 35. -o, -6. 37. 5.10 p.m. 38. a±i. 39. -^. ^. a-f 6 a 0+6 o+6 ANSWERS 389 Page 277 (oorUintied) 32 43. -^±^V5I, -|±iVl3. 44. 10 in. from a comer. 46. 27. 40. 6076 nearly. 41. -»>^^^+64. ^g. 3. Page 281 7. (6. 2). (-V. ¥)• 8. (3, 2), (J^, ^f). 9. (6, 4). (-f^, -ff ). 10. (4, j.), i-l -^). 11. (2, 1), (-5, -^). 12. (2-525, -175), (-2-275, -1-425). 19. 4-196, 4-732. 20. (-2, -1), (-1, k)- 8. (2, 1), Page 283 49±21\/5 -7 + 3\/5\ 2 2/ 7. (4, 3), (-1, -3),(3^(-I, _§). 8. (1. 2). (-5, -10). {-'-/'', =^^y 9. (2, 4), (3, 3), (2, -3), (-3, -3). Page 284 / 23 II \ 12. (±3.:p^),(±^,±^^j. 13.(±4.±l).(±13V,V±5^/^). 14. (±1. ±2). (±^, +^). 15. (0. 0). (1. 1). (^. A). 16. ^±6-32, ±316). 17. 35. »)• ALGEBRA Page 286 16. (±2, ±1), (±1, ±2). 17. (±3, ±2), (±2, ±3). 18. (±2, ±1), (±1, ±2), (±'\/^l, +2\/^), (±2V^, tV=1). 19. (5,2),(-i, -i). 20. (6,2),(-2. -6),(|±^\/57)(-i±|\/57). 21 . (5, 3), (3, 5), (6, 2), (2, 6). 39. 7-32, -68. 40. ^b+^Vb^-l6a. Page 291 18. (2, 2), (2, 1), (1, 2), (2±\/2, 2 + V2), (3±i\/=7, i+iA/=7). 27. (±10, ±5), (±5V2, ±5V'2). 36. (3, 2), (2, 3), (-2±V^. -2q:V^). 39. (-1, 2), (2, -1), {-it^vTS, -i±|\/T3). 40. ^+^^'-^'', -s+V2d^-s\ 41, (4^ 1)^ (2, 2), (J. 12), (f, 6). 42. (4, 2), (2, 4), (8, 1). 43. (5, 1), (1, 5). 19.81. 21. (-^-.^). 23 be Page 296 19. — . 20. x*"***. 21. 1. 22. 1. 23. a\ 24. 1. 25. 22", 312. 26. 3 27. 2, 9. 28. 2, 7, 3, 2. Page 301 45. J. 46. 8. 47. 625. 48. 11 J. 49. 125. 50. J. 51. Tig. 52. 32. 53. 4. 54. §. 55. ^. 56. J|. 57. ^. 58. -^"53. 59. 16^ g^ gi, 1^ ^. ANSWERS 391 Page 302 1. x-\-xi-6. 2. x2_i. 3. a;2_i. 4. 3a;2_8x2+9x-10xi 5. 0—1+40-^—40-1. 6. a2— 2a-^ + 3a— 2a^+l- 7. a;i+4x-lla;^-6a:^. 8. a:2+8a;^+24x+32a;2 + 16. 9. x^+xy+y'^. 10. a'^— 3a+3a^— 1. 20. x^+x-^-l+x'^+x'^ 21. 5a^"+4a"'-2. 22. 2x2+6a:+2. 23. l-2a. 24. a;-2Vz+3. Page 304 I. x-4, a3-63. 2. a+l+a-i. 3. x^-2x^-x+2xi+l, 4a2-8a+4o-i+o-2. 4. a2+3a+3aJ+l, 1— 3x5+3x-a;i 5. x*+xy + !/*. 6. x^+yK 7. a^-\-b^+ck 8. (x+2/)(x2— y2)(x^+2/5), i 9. J-bk 10. ?^, a^+oM+fti a-Vod+fc. x2— 3 II. x^-2, x-x^ + 1. 12. 2x-2+3x-i, X+2-X-1. Page 305 3. 5, fo, 49. 4. 4, J^, 25, 4, ^, 8, ,^5. 6. S^g, 2. 7. 3-162, 1-778, 1-333, 5-62. 8. 4. 9. 1-732, 10. 9f, -lif. 11. 100. 12. 4, 2. 13. i, 1^. 14. I, f. 15. ^^, 3xi+2, ^p^- . 16. x^y^Sx^y^ Vo+4 o26— 62 an ALGEBRA Page 305 {continited) 17. x^y-^ + 1+x sy 18. z^—y~\ a^— l+a"i 19. o'"'+a2"'6"»+a"'62"._|_b3m 20. t/+2?/^ + l. 21. *— 2— a;-V 22. ia'-ift". 24. -OOie, 1-44, 3-375, 8. 25. Vs -lY i/'o + l 26. 1. 27. e'+e-', x^— 2a:2/^ + 3«^y— yl M. a6c 30. 4, 32. 31. 2750. 32. ^. 33. 2, 3 34. x^ +2x^ + l + 2z~^ +x'^. 36. 2z+x,^-a;"i. Page 310 7. >^4, ^27; V^16, ^27"; ^64, ^8l, ^125. 8. 3a/2, 5\/6, Vo, 1-26, ^5. 9. 12\/2. 10. 12V5. 11. 33\/2. 12. 3^2. 13. 7-^12. 14. 10v^2. 15. 9V3. 16. 0. 17. VS, V5, xV^, r^ix, V2. 18. 2-52, 3-78, 126. -63, 126, 1-26. Page 311 25. 3\/iO. 26. i\/5. 27. ^VE. 28. V2. 29. U2V2-V3). 30. ^02+6*4-6. 31. Va+b-Vc. 32. -(z-Vx«-y2). 33. 2-517, 1-364. 34. 194. 36. 37(V3-V2). 36. M(7-V6), 2V5. 37. i\(18-34V5). 39. 10 ft. 6 in. ANSWERS 393 Page 314 16. No root. 17. 4. 18. f|. 19. 100. 20. 9. 21. a. 22. 25. 23. 64. 24. No root. 25. 5^1^ 2a— 6 26. 3. 27. 10. 28. 10. 29. |. 30. J^ • C2-1-1 Page 317 18. (4, 9), (9, 4). 19. (4, 16), (16, 4). 20.(17,8). 21. (9, 1), (1, 9). 22. (2, I), (|, 2). 23. 2, 1. 24. 7, -6. 25. (2, 8), (8, 2). Page 320 10. 2-823. 11. -196. 12. 2^(2+^3), 5i(\/5+l), 3^(2- V3),2i(5\/2+3). 14. 1+VS. 15. ^, ^. ^. 17. 2-309. 18. 2+3V2. V2 V2 VS Page 322 ■•0. 25\/^. , 11. 68. 12. -25. 13. 1+V^. 14. -l-y^. i5_ 2a2-262. 2 Page 324 I. 2+2v^-2\/3. 2. 2+|\/6. 4. 1-98, 3-15, 1-39, 3-55. 5. 9. 6. IJ. 7. 7i. 8. 7, -1. 9. 20. 10. 13. II. 6-f2vl5, x^+2xy+y^-Ax-'iy. 12. |V3. 13. 4a4-2\/4a2-6. 14. ^(i'*-2p?+g2). 15. 12. 3M ALGEBRA Page 324 {continued) 16. x*-lx^+2x+2. 17. 0. 18. 2a, 4.a^-2, 8a^-i 2m, 19. ^O- V'. V».«-n»- 21. 5. 22. ^+l.Va+I+Va-l. 23. 4, -7. 24. 40. 25. 2-62, -38. 26. ?Vo*-62. 29. 1. 30. l(h/2. 31. 16+9\/3. Page 330 17. x2— 2wx+m2— n2=0. 18. a;^— 4aa;+4a2— ft^^O. 19. x2-6a: + 6=0. 20. 16a;2+8a;-63=0. 21. x3-28x-48=0. 22. 24a;3-26a;2-(-9x— 1=0. 24. 4x2-28a;+45-=0. 25. a;2-7a;+12=0. 26. a;2{a2_62)-2a;(a2+62)_|.a2_ft2^0, 4a;2-16a;+9=0. 27. 9, 7; a-\-b, 0; 2p, pq; 2c-2a-2b, a^+b^-c^. 28. 0, 5, -1, -4. 29. 4, 8, 30. ±16. Page 334 I. is, 6i, 22. 2. 47, -15. 3. -2§, IJ, 6J. 4, _5g, i_3a. 5. a;2_i82;+80=0. 6. x2-5x+4=0, x2+5x+5==0. 7. 6x2-a;-2=0, gx2-px+l=0. 8. 6x2-2x4-3=0, 15x2+26x+15=0, 9x2+26x+25=0. 9. a^-2b, 3a6-a3. 10. x^-xip^+2q)+q^=0. II. ax2-x(2aA-6)+oA2-6A+c=0. 12. x2-4x-4=0. 14. x*+6*+8=0. 16. i(6*-ac)(6*-3ac). ANSWERS 396 Page 338 6. Rational. 7. Real and irrational. 8. Imaginary 9. Real and equal. 10. Rational. 11. Real. 13. 4. 14. ±5. 16. -i. 19. -• 20. 2, -?. tn Page 340 8. (x-{-2+Vl){x+2-Vl). 9. 16. 10. ±6a. 11. (x-3 + 2V5)(a;-3-2\/5). 12. (3x-4t/)(3a;+42/){4x-32/)(4z+32/); ±|, ±J. 13. 174; (8a;+7)(15x-4) 14. 62^4a«. Page 341 4. 6, ^, |, 4, -14. 5. f, A. 6. 16x2-40x+21=0. 7. 12^, 8. 6x2-19a;+15=0. 9. -25. 10. x2±12x+35=0. 11. (72x+l)(73x-l), (13x+ll)(17x-15). 12. x^-'kc+3=0. 13. 2a + 26-2c. 14. 97x2-53x-17=0. 15. (x+3+V'2)(x+3-a/2). 16. 2x*-17x=0. 17. ax^ + 3bx+9c=0. 18. acx^—x{b^—2ac)+ac=0. 19. 1, °+^~^ . 23. ±12. 26. ±4. 27. 3mn-m^ 28. a. a+ft+c 29. ^ » °t^ . 30. i 31. c+6-a. 32. o=8 or 0+6— c o+o— c 33. 8, 1. 35. 6. 2. 36. (x+b-{-Vb^^){x+b-Vb*^). SM ALGEBRA Page 345 16. 2a-6+3. M.Pllll±^. 18.-7,1,19. v—q+r 19. (x-y+3)(a;+2!/-4). 20. 2o-3, 3o-4. 21. (2a-6+c)(3o-6-c)(3a+26-2c). Page 348 13. (2a;-7/-53)(4a;2+2/24-2522+2x?/+10xz-5i/2). 14. (a+6+c+l)(a2+62+c2+2a6-ac-6c-a-6-c+l). 15. a3-63-c3-3a6c. 16. 8x3-?/3+2723+18x?/2. 17. l-a3-t3-3a6. 18. 8a3-27fe3-64-72a6. 19. l+a2+&2+a-6+a*. 20. 9w2+n2+l+3TOn+3m-». 21. a+56-1. 22. 3a+fe. 23. x2-3a;+9. 24. 2a-5. 25. a+6+c+d-l. 28. 27x3-8!/3+334.i8xj/z. 35. 0. ^® 2a-36+3c ' "l ^^- (-+2/+^)(a+6). 40. a,b. Page 351 9. («-2/)(t/-3)(a;-2). 10. {x-y){y-z)(x-z). 11. (x-?/)(y-z)(z-x). 12. (a-6){6-c){c-o)(o+6+c). 13. {a-6)(6-c)(c-a)(a+6)(6+c)(c4-a). 14. a+6+c. 15. ^, ^. 16. ?±-^ ^-:::^- 17. a, 6. 18. ^^ ^■=^. a 6 a— 6 b-\-a a b 19. *, *±^. 20. ax-a-b. a b—a ANSWERS 307 Page 353 10. x*-2x^i-4x^-Sx+l6. 11. r»-3a;2+9a;-27. 12. x^+x'>a+3^a^+x^a'^+a*. 13. (a+6f-(a+6)2+o+5-l. 14. a;— 6, a+6, X— 4, m+-, a;+y— 1. 15. o* — 1. 16. m*+l. m 17. a8-6». 20. a2+ap+3=0. 22. 1, 9. 1 II '11 ii 2 4 5.1 a2 la 4 23. a;3 4-a;3a3_|_a3^ x^— x3a3_^a3^ a;5_|_a;5a5_j_a;5a5_j_x5a5-j_o5^ 4 ai -k I la 4 a;6— x%5+x5o6— ar^a^+os. Page 357 13. 3(a*4-6^+c'^)-2(a6+6c+ca). 14. 2(a2+62+c«). 15. 0. 16. -3(a26-afe2_^62(,_jc2+c2a-ca2). 17. {x-y){y-z)(x-z). 18. (j;-r/)(^-2)(x-2). 19. (a+6)(6+c)(c+a). 20. 24a6c. 21. 3(x-?/)(j/-2)(z-x). 22. {a+b){b+c)(c-^a). 23. — (a-fe)(6-c)(c— a)(a+6+c). 24. {a-b){b-c)(c-a){a+b){b+c){c-\-a). 25. 1. 26. 1. 27. 1. 28. -4-- 29. 0. 30. a+b+c. 31. -{x+y+z). abc 32. 3. 33. 6abc. 35. 3(x2+?/2+z2) + 2(xy+2/z+zx). Page 362 24. 18. 26. 02-262, a^-Sab^. 27. a«=c*+26«. 28. o»+2c»=3a6«. 808 ALGEBRA Page 362 e. k^+ri\y^-^iZ»-{-kxvz. 9. x=(a-l)(l-6). H- ^ aoo 12. l(a+b+c). 15. -{ab+bc+ca). 16. ~- 18. -i(a+6+c). ooc 22. {x-\){3^-2x-^+2xi2). 23. -a-6-c. 26. (x+l)(a+6+c). 29. 3(a:*+i/2^z2) + |(a;j/+j/2+2x). 30. o2=26+c. 31. 9,17. 32. 38, 70. HR 3—8-05 .( 4^£'J4 . \ -- \ a -- B ■ Mi hM f8ei7 sot^eo t72t76 s., Auvusn o a dO xxisuaAiNO CURRKUUM LABORATORY u./ lO : \9^ LIBRARY USE ONLY ,1111 f university of - british Columbia ET-6